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Muficalibrisduatuor ^ \ 

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P ARI SIISJ 

Apud Gulielmam Caaellat ^in pingdl 

^Gallina,exa<iaer(b coll egii. 

Cameracenfis. 

1 f / z. 

Cum priiiilcgi'o. 



irfnotta 




GVLIELMVS CAVELLAT BiBLIO 
pola,Candido'Lct^Qri S. 

Ntcrfui multorum coUoquiis candidc Ie<5lor>qui mi* 
rabantur matbcniaticas rcliquat ftudiof& cou,vm* 
cam muiice deferi, Arquafi inuriJem,aoc iUibcralctxt 

iaccrc : qux apud veteres tantum {hidi| acveneraticv 

nis crat affequuca, vtneraocitrahuiusc^ninoncm Uoslibaa^ "^ 
liter eruditus habcrctur. Audio Plato^eni acreliquos authorcs 
cla(Vico$,vix ac ne vrfquidem ab eoaui muficam non didiciflct 
mtclligi.Idquod vcl vnum cxcitarccebucrat omnium amc«^m 
dcftudiummrcm tarn ncccflariam, NondubitoabHuqui fcrio 
Philofophantur rctincri 8c habcri magnoin preao:fcdcam ficut 
rcliquas mathematicas publicc doccri non video. QuidcaufliiE 
cft?an arcana magiscffcdd>ctrcliquis^anpauciraultis inuidc* 
re tantum bonum dcbcndlntcUigo Latinorum muiicorum ma-i 
^ . gnam penunam^eorupr^fertim qui artcm caliuerint,& mctho* 

L do I'cripfcnnt , cxemplaria non adeo multa & ea non fatis anen* 

data,quf caula vidcFi poteft,cur muiicc minus iitfrequens.Qua* 
re cum proucntum mathcmaticaruta plurimum delidercm. 
Offrro tibi clementa muficalacohi Fabrrin hoc gencrc fcribcn- 
di vt audio pr^ ftanrifsimi: que obi gratifsima ac vtilifsima fore 
confido,fi nulla fit doclrina melior,quim qugperdemenu tra* 
ditur. vale acfru ere. 

V"^ 'Ofc. 



I 



t X 

KicpUum ic viaqumUeitnqwfttommVraftitnum. 

If acyjii*^ ^mucrAm^cl^iffm^ uhyftuUimids ilcntentoruvt wuficdUuWi^udi 
m Ci|i^Xif/nmgftg funt^ friusMcdre aigdids tqukmfrchdtdi ccgnotdifjim: 
R IJmH J f^^Mf i<i/ff< minmic hteret^cmcmfirdltmesy in quibus ud folis uis 

•■■■■•^■■*cM|ww ducis, nfutdfqueihimcxdntiquisfumntcpere muficos 

ctmmendd^c4^Acres utrt udlAunt. 

Mer cMrii4s, Tdmytds. vtolemats, 

Orfhcus. Hifmrnids Thebanus, tjtboiides. "^ 

^yths^OTds Sdmius. Ter^dnder Lcshius . iUppdfuu 

Amf^ioHThchanus. Lycdon Sdmius . At^cxenus. 

l^JHtis. PfOfhrdftus Penotes. PhiUHdUS Vytbdgoricus, 

Anon Le^bius. EjtiacusColophomus. Architds TaretiUnus. 

Myddsrhrygius, Timcthcus uilefius . AJldms, 

Carcbus lydius, Nuomachus^ viyus Seumpus Bpaiuj , 

nydgnis Vbryx. pLtf o. 



fJidrfids. Ariftotdcs. - 

^t fimUmmqudmpiurimltquos omnes ctentd mmoria difdflindrum cbUndittfjU 
mdMitficd reddidit mfipes. inter quos^ duos fraceftcres meosidc<Aum HdU- 
^iam^CT^ocebum TurMinum dnnumeroztdnqudmeddrtefoJleritdtiuiSur^s, 
c^tmimenidiAO' earn hwr^ci eius effe&us, vytbdgorid emm dnimorum ferocidtn 
tibUs fidikifque emoUiAdiit. Ajcle^da frementis uulgji fediticnes^ crebro cantu 
comj^fcMtMimque tubdfitrdis medebdtur. Vdmon Vytbdgoncm, ebrios crfro* 
inde fetuUntes ddd^ctntes, grduioribus modulisj dd tanper<Mtidm reduxit. u * 
brem tT uulmra,mupcd maduUHone curduit antii^uitds^i^em quoque fuduttdte 
Ifchiddis fCoxendicimque dolcres emendduit'^^juod aifmetdds Thebdnus tentdffe 
memcfdtur. Theofhra^, di dnim ferturbdtiones moderdndds , muficos ddhi-- 
buiffe mcmcTdtur tnodulas.i^ec inmidqmdem. e/1 enimMuficdjUt qu^ddm mo- 
dnationis lex dtque reguld. Qudmcbrem bono iureeos ridibdt Viogenes Viuficos^ 
qui cum cithardm dd barmonicos confenfus hdberent temperdtdtn , dnimum gere^ 
rent inc<>mfQfitum,CT prorfus harmomd uit4t deftitutum. Xenocrdtes , orgdmcis 
modulisylympbdticos liberduit,TaUsCretenftsJuduitdte cithdne^mod^osjfeftilen^ 
tiiimque fugduit^ TerfdndcTj cr Anow^Aowff, cr l-wtioj cdntu kgrduiffmis 
m^rbis Uudfje^diuus Scjerinus dutorejl^ HerofUlusmedicuSf egrorumuenas mu- 

A i/ ficis 



ML;--t 



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ruistmUuU>^numnh rimAcusautcmficus,4umuoltiit t^trdfwufwJJj* 

fitxitJbec tfi, faiMtt Ummm mnti Ufju - d tUksum ctmtf *i moitraalt. 
m cjpihfahorA:a»lbtTtuoia>us Miumwr.^^haut i»*»»x«iM Wed. 

(ilMftt Anon. SCTPrttf«:f«oirJ*w rnwf wrw. iff »lfor*» m4»f j; c*I»»/ i^,^ 
f<oi»r.iii aa\4t»mmtm4crec\tharAm.ftrfmartmmn*U(t. m<j«i*i dflxirfl* 

£ «b tUnSmtM MKMt cmmtniaiimtm *iAucat.dt M,Utittpi»ttrrt* 
cntum ^pn/twiw cmmaOAUm «Mfc«ri»-tJtw : (iti nwrc «& Nk -»^« ** 

lucm hJiitmMt.U me mm inter tuoi ctitntuloi tuarum nirtutm^uiqut »o»»«: 
nUohfenkOmmrffecoffiofdta. vdt^ 

I 

eucchus stawdenfts ucohc Lahinio CT ^^ctho Turhdmo Viuftdsz 
fuis chariHimu pr«c«pfohi^/ * 

yod McT ofdtifrcm atquc rhetcrd : id intncjntcrem C tnuficuM 

intcrcffc uolunt. ncquc oratonm qumquam Set mrcri : quiiitm 

, rfxtor ncnfit. ltd f^jhumftrnfaludiciumfuit: utnc c^torqui'^ 

dcm did matatur unquam/iui idem mttftMi non fucnty fulcbfc 

iiii, I miwoj ,cr bi/friowfi d cdntorum hcn<fio cem ftquejirantcs tan- 

auAM ificurcoi i ptrw maifd > cjJlSque fhil0fofharum dogmate . ntc mark, 

nam nomcrus diuinus foctd ubiqut doetum vyrix moduljtum introduat caH^ 

tmm : utapd odyfftm ubiPendofen^VhamumfididncmadntrUosunburnii* 

ucntemhis ucrbhdU^utdm effingit. 

Vf^UT€AilUchrimdTu tUuumeJl affatdcunnftem. 

vhcfmmltdtmcihominummukoHidf^^us 

load hcminum dtquc deumizi ^}i^\dud€m du&ari^m dddunt. 

tx Hi fdftgt dliquid, f I N f. - 

U quda Homaus poti/.ww nnnwif tdlcs efft duhito,ut qui dudum duobufrt^ 

md mufxcti rudimcntd pncipnim.CiuafroftcrAd uosnoftros Ubom cxjiminan^^ 

doscommkto.quos co [uKntius mefulcepilfcfdtccnquo ynufiCdUm faatt^^ 

quedfudsr^coi ncqucldtin^ quidm unqudmtUmmis trdmmejfe kgtrim. 




V.W-.l.'JL" ""'Vi^^'"^ V^'*3^'»lt'!|Kf 



^ Jf 3 

dt introduiliims quamflurlmds. inter quds ed mmum nchlifptrnd tfl^qudm di 
uHs scuahius BoettusfuimomatHniiim reliquk : qunn tmum in hdc re fr^ejertim 
ddfff mcofum fiudi^rum iuctm. si ergo frolHmtrifisifttis wihi eft. in re enim n0^ 
ftrd malumusd!krumiuiicUfeqmyiukmj^fridfrobdre^ Vdlete. 



■^T^ 



AHtiqui quidemuficd fcrifferunt 




inftgnes cxrtftnimikiis 
qui detddeikfcrip fere. 


Demoiritus 




?ldto 


Albmus 


nerdcUdesj^tUus 




Ariftoteies 


UuusSeumnus 


rtmofheus uikftus 




Thcophrdftus 


B4/iUM5 


vhMdus FyAagoricus 


nkomachus 


uiUms 


ArcbUds TdtfOitmus 
x>tt0Tbeodm 




Artfioxenus 
Vtolomeus 


1^ 


xdntbus Atbemenfts 






GA4ftUS. 



Ar^umentum qudtuor librorum mufices. 

Cvrimus liber interualld mufids moiuldtl^ibus dccommoda difcutit. multiphx: 

duvUreftrifUre^ quddrufltre, fuperfdrticularetfifqudlterurfiyjcfqHitertium^fifi 

qutoR^mumJfiiffjciuiQ^aukmfUrfefquio^dUkmfqudterJefquio^^aUu^ 

f;^ fexiesfefqmUUuuM^ 

C,secm$dui:de tcno^integro tonidimidioJimUcnio mincre^emitordo maicre^com^ 

matefjhhifmdtc^atque dlafchifmdte, 

4LTcrnys:def€fqMtoriO,ditono,didteffdrcnydiafentejdiafent€&t<m0idid^^ 
diafafoncr^i^^^^^^<> > didfdfon cr ditoHP, ^pdfon cr diateffuron , diafafin 
Cr didfente, diapdjon di^fentecr tonOfdifdidfaJon, ac tntegro toni & confonan* 
tiarum omnium dirmdio. it demdxmartimhdrmomdrimiconfindntiis^ &qud' 
rumddmmediiXAtum. 

^Quartus de monoihtftd^iUtrddiCYio, fentdchordo^heptdcherdo^ oSoehcrdo, 
fentadechordo,didt^dSi^hfomd$i€iSy(narmaniiis melcdiis.it dt mlodidrum m^ 
du-CT h^cjub brtmdteeontjtddaidrffmmtmn Ubri Junto. 



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-Ci^coti fdin Std^Unfu ekmetHtrHfn rmficdmm di \ 
cUnfftmum mum nkoUtum ic HaqutuilU 

Tf4tfiimm Tofificnfcm \ 

liber frimuu \ 

n;ntnu^!Umj(jifi^Srduis^cuUjtie fffdciomn hjbitu^c. 

hsvdcmm uoc4}nus ncruuntj cbomiw, exputa^ amm, c quia 

iqu^fvniUcjijd quo fonum cikinrns. 

\uiu\xivUx micrujilum , t^ quotUi nuius fpdctum co^tinet tm* 

11""' flmies. ut bisy C J«p^* intcmillum didtur, ut tti^ CT di- 
cirMf frlpffX.«r-j¥4rra,5r^«^»'''p'f^ nuncu^dtw.a-^oc pd^o dchtceps, 
jnteruJUfH fii^rpdrtuulxrc ^ ejl cum mdui continct mnus , CT psrtm tlus 
dUqudm. ,^ 

VdrSfCd qu^c metitur Mum^pfufn diquotUs futrtfU reflituens. 
Si maius continct minus CT ciufdcr?^ dirmdis^m , fefqudtcrum nomindtur in- 
tcruaUum. 

5i rkuus continct minus cr minaris pdrtm tfrtismidicitur fcfquitcrtium. 
&t ucro continct minus tX ciufdcm minms f^cm Qilduam : cpo^domn ypfqul- 
o/faKu w^M^ nominatur. it itd reliqud fufcrfdrticuUrid intarudlU, fer fddle cfl 
^di^wreijM fdmm rmftco coniucunt profoftto. 

SupcTfdticni dutem dicrrftur intnrudlUm,fi tnMus contincrct minus C mnoris 
f dries dUquctyqua fartm unam non efficiunt. 
fiumerifunt ddinuiccm frimi^quosfoUmctituT unitds, 

TDignttdtes. 

,\tcqmd mrtitUT dlterum,metiturCrofmc menfuYdttm db tCtf . V 

IcompofituWyin cd rcfoUdturfimflkidyCX quibus compouitur. z 

fcMiuflwct numtri fotsi^ututds^ to dtnomindta. 3 

I VKit^ ui quemcunQUt numerum itUbtJipfum frrdudt. 4 

tduicqidi metitur detrdOum gr re{idm(m:metitur cr totum. 5 

Qujtatnque uni cr ddem sequdpntyCT td inter fe funt ^ud. 6 

ttqu^e inter fefunt^H4 eiufdemfffnt ^quemultipUcid,dut fuh multxpUad. -j 

lorumquortmtoUJtmt0equd,tr^''^^/^^^^^'*' ^ ^r n ^ 

Quicqmd bis duawHdhqmi trof^ceniitM ultrd iUius dim^dium t[]e w w/Jc ejl. ^ 

U quod duplicdtum non mflet inteffruWyid profeClo non continct dimidium. j ^ 

Omnc totum rff mms fU fdrte. 11 ^ 




I v<ttdaqyalichcrdd,qUiefp4cijadffddumfrof^ioeftyedme^^ 
z inter mmmsf^d umtdtc diJianUs^uJhim intcrdfi^offcmeixum. 

3 Sfdcium ^uoiiihct ^in quotlihtt aquas tdftes diulitre. 

4 Totum ddfuamfartcm,crla:9usddfete9jfum^duionwfinuniedcrf. 
y omnem frofortionm effe tdnqudm nvv;m dd numerum. 

6 si yiumerus numtrum datum wultifWcet , \demque frcdu£iuw diutddt^ numnm 
ddtumudirc, 

7 si numtrim datum mmirus diuiddt^tr quod frcumtYurfuswultifhcet'. mmt- 
rum datum rcdne, 

8 txtremorum po^ortiomm , ex mcdiorum ffofortionibus utfiU fdrtlbus ,([fc 
comfofitam, 

ixtremi ordindt:irum fro^ortmum : mdx'mus , minmufquc ttrminus inttBh 
^untur. 

CScientia fubalrcrnata^qualis ad arithmeticam mufica eft , principiis 
6c dcmonftratis fcicntixprioris fubaJtcrnantiTquc vtitur.at ftuduimus 
vr quam fieri potcft rarifsim^ in hac difciplina fiat: vcrum plcrumque 
vbi,oportiinum videbitur^alio xjuaminarithraeticis fafturacftvte- 
mur demonftrandi modo: quo fingula magis ex propriis faftavi* 
deanrur; quamuis arithmctico fuffragio atque lis qux in aritbmcticis 
nionft^ara funr^eadcni promptius faaliiifqucficrent.attamenquando 
idficr &quado non,fcquentibus dcmonftrationibus cuilibetp^rquam 
perfpicuum euadere poteritk 

I €si intaudUum multifkx bindrio multipUcetUTyid quodfit ex hdc 

multiflicatlotjeintcruallum multiflex (Jl. 



i6 



Multi plex exduplatione. 



I A I B 



I Mulci plex interuallum 



C Interiiallum binario multiplicare , eft interualli habitudi- 
nem duplire : qux quo padlo diiplanda fit , dudum tertia pro * 
pofitionc quinti arithmetices notum cffc poteft . vt fi a b fit 

A iiij inrcrualluia 



I 



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iotcruallam quodcunque quod birurio multiplicarc iubcamur : ipfum 
pcrtcrtuun qumtimulripUcabimus fi ducimus liP fc & b in fcprouc- 
niiotquc d c critquc produaorum d & c duplex intcruallura iarcruaU 
lo produccntiutn a b.Quia rurfuna dudraus a in bproucniitqucpro- 
du<a-um CI eric per fcxtara quarri quse proporrio d ad c cadcm « ^dc 
& intcruallum d c in duo cquapartinira.Scd cfto nunc vt inrcraalluoi 
a b fit multiplex qucadmodumproponicpropofinoquod modopaulo 
anccdiao,binariomult!pliccrur,fitqucduplura d c & ca proporno d 
adc & c ad adico mrcruallumd adc clTc multiplex. Nam turn a ad 
b multiplex fit intcruallura.cum gcminaturmamfeftum eft tflliltipler; 
adii multiplici.quarc per viccfimam fecundam noni arithmcrices fta# 
rim notura eft inrerualluracompoficum multiplex cflc.&propofitum. 
Sed ide aliter hie demonftrarur.Quonii cnim qux propomoaad b ea 
eft d ad e & e ad cnam vcraquc proportionis duplate a ad b medic« 
tas.Proportioautcmaad b pofita eft multiplex : igiturfic proportio d 
ade muUiplclc eft.raetirurifjturper diffimnoncmulriplicis: e ipfum 
d y el bis vel ter vcl dcioceps. Et eodem quoque iurc c totics metitur 
ipfum e.quarc & c per primum communem conccptum eriam mecitui* 
ipfiim d.Quicquid enim alterum mcnturrmctirur 6C omne quod men^ 
- luracura eft ab lUo.cft itaque per difFmitionem intcruallum d c muU 
tiplex.quod erat oftcndendum. 

est fiurmt termini ^oporttPHjlitfrcatiftitutixcum f riww/Wnf ultimo 
compjrdtusji primiw uUimuvtfuait matfui ^ctictur ^ficumiMm, 



^4|tf| | i|x |4|A b c trcs termiQ! ^por 
I^T; I c I |d I c I f I lionalesquoffinueratb. 



CTmcrfi banc dccimatcrtia quarti dcmonftret fuffidenteneam ta« 
men amplius hoc m loco volumus manifcftarc. Sine cnim a b c tcrmi* 
ni proporrionaliter conftiruti 6C a numerer c.dico idem a numcrarc b- 
capio enim d e f cerminos m'ca proportione minimos. Quoniam enim 
d ad e vc aad b:& ead f vt b adcigiturperequam proportionalita- 
tern quam ricefimaprima fccundi arithmcrices monftrat:d ad f y t a ad 
afcd a per hypothcfim mentur ciigitur & d mctietur f. At quia d e f 
pofiti funt in liu proportione mimmi ; ergo per quintam quarti d 3c f 
liint admuiccm primi.Et cum d fcipfum metiatur panter & frpcr dif- 

finitionem 



« f 

fttiltlontm d eft vnltat. ANrero cumvolUJ fltomnUnumerli pars: d 
ergo mccicrur c Et cum a ad b vt d ad crmctictur igitur a ipfum b fc* 
cundum cermlDum^quodcraccognofcendumacquepropoficum. 

CSi intcruallum binario multiplicatum, multiplex cfFcccrit intcrual- 
} lum:ipfum quoquc multiplex erit. 

\ X I 4 1 S I A b interuallum bmario mumpii- 
i a 'n b I c I catum multiplex cfficiens. 

Ch^c cji conuafs frimtticuius h^ecefl ration am cum interuallum compofitum 
multiflcxjit cr ferjecundam ccmmumm fcimtutn refoluatur in id inuruallum 
tx CUIUS multipUcationc (criuit Jiquidcm comfofttum omnc : in eafmplicia dijfol' 
uitur exquibus cQalitum,concrecum^ccmfojttumqu€ejfec0^nQfcitur)^erfcxag€^ 
fimam nominterUAllMyn fimfkx mu^ifUx crit. At id dcmaliteroJlcndUur. Waw 
ji.a b c ^emindtum interuallum jit multipUxfic ut c multiplex fit ad a 0* ea fr^s 
fortio a ad bo* bad ciquomam c adaent multiplex, a fer diffinitionm metier 
tur iffumc.quare cr f^r pr^ccdentem a metietur ipfum b.efi iff tur interuallum 
fimpUxb ad a per dtffinitiottcm multipUx.cr propopcum. 

f;Si terminorum intcrualli primus ad fccundum comparatus , fefc vt 
4 tertius adquartumhabucrit.quot proportionaiitcr mcdijprimo & fc, 
cuDdo,cbtidcm tertio He quarto mrcrucnircneceH'e eft. 

If mediu propartionaie inter d ^f ttrtiu Q- quartuiut t? mttr primu Q-fec uiu 
i:x7 Iit>| ix| |it^| IX I g I I 5> I 6 I 4 I I X I 

fA ( b I C I I d I e I f 1 I g I h I fc I I 1 [ 



^Hanc duodecima qudrti monjhat. cuids adhuc Inc ut adfequentia momentunt 

habens icognitio promptun habeatur,rep€titur dmonjlratio. 

Ciir a cuiufcunque inter ualli primus terminus ad c fecundum, utdt^ius ad f 

JuartumiCrp^ b medius proportionaliterconftitutus inter a cr c.dico etiam inter 
0* f inter uaixre unum proportionaiitcr medium. Capio enim perfextam quarti 
g b k trcs minimos fccundum proponionem a ad b: CT argumentor ex uicejimd' 
prima fee undiver <equam proportionalitatem^ad but a ad b ^ h ad ^ ut b ad 
cjgitur^ ad k utaad equate CT ^^ <i ^d f. at ft d CT f ft^utiidem termini ctwt 
J iiimanileftumia/fi efi mter d cr/ mteruehije unmn proportionalker medium. 



B 


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I 



lalitcrmtJuu tnminUd crf.<l''<^'ratdmo»Jh<vtium. 

CSupcrparticulari$iaterual!imcdiusnamenu:ncquernus ncqueplu^ f 
rci proporrionaliter intcrucnicnr. 

I 17 I I T" 1 r llJlA^ paaUni luperpart.cularc. . 

] A I K I g I Id Icl Tifj Dcttresmimmiproportioiiaadb. 

«•« m«f «To d cr f adtnuum frirm : cums offcfttum urn monfirutum tft CT 

(olLsMmtatis dif^imnc d,fl<nt.mox w^mtntor d^^Jutg ^df. f<d v,urd Cr 
i vof,tHS eji uJs int<rua,irc medium, igftur fn fr^cdrntcm cr v,tnr ^ Cr f »«• 
i^oi fcU umiX4te difianUs wuruimt innrcifiturqut aUq^ts nwimus wtdiw. 
quod pftfccundam vctitioT^tm ifi in-fclpAiU.ncn ^ptur fu{erfartKuUrts tnttr- 
Llii InusmtSusintnUi'HU^iumcTus .mquequocjucccdemar^uvicnto mfim. 
mentflurts. it banc rtum d<m>nPr*tffxas^fmafrvm *<m:. tr ex bur qticqut 
c0rm>Uur:T(pnha>f,onem dmanfh<Ui*nu Ar(bu*,^«id rm\Ufufrri[mxcuuns 
induo^UA dwidi po/Jit quam ttrtsc ir.jhtuti(mu mufiaec^pte undtcimo utdc- 
m diumStutTimi Adutnt^on temtffcdiuiscMtriHi quammhem ahi^utd 



lioTMn 



X 6 

borum quds lHterfTemtitrclt^4Uti{ftmi,<itdO'f^^c ^ fUr{fqu€ turn arithmHic*^ 
tim% nuifictt iaflUutionis iocis fcciffe comfcritur* Et non mrentur <juid4m (fac^ 
omnium dix€rim)duslibros ferlujirarcncn utfdantfed »t txfurcuUnt ^ C4ni^ 
no more fi quid hUnSyfubfultdnfjuc dffdrcat : altius dentem infigdnt, tnotdcant 
CT" rrprcbfwdawt O'fkniiHm oiuff^tralnlcvUlofophiae nomeu ^fuorum ccnui* 
ais UccrentyO* qu^d altori4m mcium eft(fi quod efi) diui scucrini cxcltmcnt crro* 
rtm fudamque aliorum criffidnertum. cr cum »M norint^uolmtt littcrdrios du^ 
CCS infc6}4ndo, carpendo, lanUndo omnU fare uidfri : atquefuprd 1?hilof>fhic9S 
nertices fcjfgjioriabuMios extolUre^fdciuntqufquod efi afud Qormcum ne xntellu 
gendo ut nihil intelli^dnt qua idemflatim uc maledkeredejindnt amm^net : ww- 
ief^Cbt ne co^nofc4Mf Jud. sed h<tc prater pr^fcnt'is negQcijofficium.de vhilofot 
fhis enmtqui reiiefucrit nature Uijhtuti : nel abfque monitionefemfcr rede co* 
gitdhwHt.qui enimfccus fdciwntiab eorum fc fegregatU ccnfirtiOi^ ut Pbi/ojopfjf, 
qudcsfe uidaiuolMiftjnondtnplius effe cognojcantur neceffe eft. 

^ CSi intcruallum non multiplex binario mulciplicctur : id quod fit ex 
hacmulciplicationencque multiplex e(l>nequc TuperparticuUre. 



I p I I 4 I A c interuallu ncquc multiplex ncquc fupcrparticularc. 
|Aj t> I c I A c duplatiiinterualluabquodquidccftnomulnplex. 

e^sitintcruaUum nonnu4ltiflexdb:^duflumfttd c.itautcdfit froportioddd 
b^b dd'^idicQ iuterudUum d c neque midtiplex effe nequefuferpdYticuUre. quo^ 
mjmfi d c prhno ponatur nmUiplex : ergo per tertidm prsefentis cr << b interudUi 
exit nudtiplex .at pofxtum eft no multipUxJUfi fea^ndo dixeris d c effe interudRum 
fi$pcrp^rtuuUre , cum fitdddb ut Jf dd cierit interuallifuptrparticuldrts mcdius 
proporttoHdUs terminus, quod per pTdecedentem eft impoffibile.eft itzique notum ft 
inter udllum non nudtiplex biadrio muul^lketur: compofttum interualium mini* 
medut nmltiplex dut fuperpdrttculdxe effi CT propofttum. 

- CSi iorcruallum binario multiplicctur,atquc id quod ex ea multiplica 
tionc crcabirur multiplex noa litnpfum quoque nonerit multiplex. 

e.H<£ceftconuerfd pTiiecedentis. sit ergo fuperioris prcpoptionU figurd in qud in* 
tcrudllum d c prouenidt ex dupUtione hdbitudims interudlli d b:cr non fit 4 c ins 
terudUum nu^kiplex. dico itidem inter Udllum d b non effe muUivlex. Hdrnft dbin* 
terudllu nudtpUx eft:cum dcper hypotbepm ex interudUo d b kndrio multiplicdto 
fitr^dtiergo per prAnam pnefentU interualium dc multiplex erit.dtpcptu eft non 

B if multiplex. 



p 



jr 



trit (juod crat monjhdnium. 

CA numero partiumfuperparticularis interualli.quxin vnumadaftM 
totum rcftituunt & numero vno maiore : quot interualli m.iorcs tcr. 
mini fimul,minonbus fimul accepris refpondeant,cognoicuntur. 



]£ x\ 3 I ^+1 5 

.1'- 



VI 4l 5 1 



Tr 81 



D . ^1 4' ^1 --L 



t>4rf<c«l4ri> 4 cowti»«m t €7 «»•»'" f'^'''^'« ^- " 'S'""' P"" /j'""''"'" "««'5' 
urn c,c«/?irM4f f«Mm tomm y, cri'f i numcrus unomaorc: duo afurnfUfc* 
cM»J««».im.rJ f,*'?"''#t/i.mpriiffci^d^m a.N4m4/-«una««c/«mpf4: 
cont,ncntbfu,*nium c fumfta zyivfu^ T^tts bfuntftasfau^dmc.Atqut 
folUumcjltaiQiH^dum c co4ttuatunumb.istur<tfuWHdumc fum^ta:com- 
intb f«w«4iici*«aum nutnaum unemMoremcAt ucro d fofitus tfl numnus 
uno Jd0TtJgodfumfUfccundumacontin€ntl>fumftafrcwidumd.quod erat 
monlh4Mdum.€ulUxr»plartm requirii dc(Urationcm,ji a ad b inUriMUum e)t 
(^Zdtmm-.quonLm afcfquAitni <Jl4d b,ngoa ccntmetb o-nuidxmdmm. 
^duo dimL fncomrmne proIoa«i«m ifuum mum uddvnt. die o trio duo a 
unti ctfequ^ntitru b.quoniJ m duo a contim duo b,cr '«/«fw due dwudta b . 
au^unu brcjhtuunr.crio duoacownmtri^b,tribuPiueeuaduntaqu^Uitfta 
Id b f/l refquitcrtm-.^ continct totum b v tfrtiim fartm bMtres tati^b^que 
runtun{h,agofuPcriorc^Sumcnto, truacontinmquatuor b^quatudrquteu^' 
dunt^qu^Jmd^t^ quMunqucfu^n^^rtitulariym^imstmnmfecwtdu pr*- 
vonio^mincrtm numerum: ^qu^tur mnortbus tcrminis [tcundum propcrtw- 
nU mdortm numerum fmn^tis. utfia^bftt Qfquiquinta: quonUm mtnamfef- 
quiquintxfunt 6 a J-iJ^ '^" qumque 4 ^quifunt fcx b.v m non mmnHs,quo. 
SUM iiz;-ioCunttammii^quiquintce,io 4 <equar,turix t.Er/i44Jfr/.t/r/- 
cu,odAua:quoLm rmnim fcfquiodaua funt 9 V S,idduc yuntumfmit 
quMtum noum b. tt quU rurfum 1 6 4a 1 7 l^fquwdnu^: idco Jcxdtam^^ux 
ju,.t 1 8 1. cr tf4 in c^ais. sedft id non modo ^4xticu\mta f(d ZT unmrfdittr 
(ontm^lari cufis.id uniuerfalita uerum (rit. fi n t 



CData quacunquc proportioiic: niaiores termini fccundum numcrir 
minorum fumpti,arqui funt minoribus Iccundum nunierum maiorum 
fumptij. 



jaSuppti I 3 |4 l5 I <?l7 I 81 j>|^m 1 1 | 3 kl 



[bcular. |i[3|4i5|6 l7l8|pliccs| i [ i |i |i| i' 



Su^plf |7| 9| 11 



tieccs|3[4| sW 



lo 



e.\tfit a^b qu^cun^ue froportWy cr d maior termlmsih ucro minor* iko a 

fumftosfecundum bimnjerum tninor(m:<:equoscffebfuniptisfic$0iis0n anumt' 

rum mdhremMdtn fer odauam j^rimi arithmnicesy quod fit ex a in b: Muum 

eft ei quod fit ex b in d,Kt ucro quod fit ex u in b: funt itfumftifecu/ndi^m b. o* [^ 

quod jit ex bind funt bfumpti fku/nduffi dyigitur djimfti fccu/ndwm b: nequdm *' 

turbjum^^tis fecwndi^md.crin quibuflibetalijs^eddem cjl danonftrdtio^ fro- 

fofitiwt. 

CCum aliquot maiorcs termini aliquot mijnoribus aequi funt; ea vni* 

us maiorumad vnum minorum reperitur proportio, qua? dc numcri 

minorum collcftorumad maiorum collcd^orum numerum. 



*■; 



iA 



^Hee (ftconmrfd^<xcedentis.Quomdm per ^itcedentcm minores colUCWfecim , % 

ium fUimeri4/m maiorem: dequifiint mdioribus ficmtdm^i numeYum wvioy urn col* | 

le£Hs,sed Humerus mdiormn eftmius tammis mdior* ergo eddem eft ftofortio | 

tmus mdk>rum dd wnum mmorumx qne Humeri minorum coUe^orum dd 7iume* ^ 

rum mdiorum coUedorum.^ itqudnqudm h^e du<£ ultmde condufiones fdcilcs 1^ 

funtiufis tdmen edrunt fdnloldtentior eft.ver frimdm mm edrum qu^nond eft I 

hahemusfi numerus maiordd minorem fefqu alter eft, ut trium ddduo, duo mdio- i 

rcs aqui frnttribus minoribuf.i.t ft mdiordd minorem pfquitertius ut 4 dd 3: \ 

tres mdiores ^qui funt qudtuor minortbus. ^tft mdior ftjquiquartus ut J dd 4: j* 

qudtuor mdiores <equ\ funt quinque min69ibus. si uero mdior fefquioddUusi ocio |- 
indiores <£quifu)it nouem minoribus.cr itd in conftmilibus, in multiflicibus duU 
ft mdior dufius r(? ut duorum dd unu/m:u/Hus maior isequus eft duobus minoyibus^ 
si triplus :u7tus mdior ^quus eft tribus.si quddruflus: t0ius,<3equus eft qudtuor.Si 
o{luplus'U^us <gqudtur ado6lo, in fupcrfdrtientibus uero conftmile eft. utft md^ 
ior (ft fiferbitertius ut 5 ad y,tres mdiores dequi^nt quinque minofibus. si fit* 

fertriqudrtus ut 7 dd ^iqudtuor mdiores aqui funt (iftem minoribus*Sfuero [u fi 

vero6luVdrtiehs ut 17 adp'.nouem mdiores, tcqui/unt decern ^T fiftemmtnori- | 

hus.O' fyoc T^^^ '^ reliquis.dx ultimd antemyhdncinfuj^erfdTtkuidrihks'utedi | 

rczuldm elicimus,si duo mdiores ^quifunt tribus minoribus,qudtUCf'fhiis, ftx ii 

B u\' nouems* ^' 



f ^ 

M 

, nouenis oHo duadenis: unus mdloru dd unumlnnufif^iualtcr eft.ltfi tns^qtd 

jut iUAtcTHii^ix oaoHUimdicrdd mtnorieft f€fquncrni4sSiquamoTmj^ts^* 
- qu.junt qtunque mmonhus dut oaodtnis: unui mdiorum umms mincnr^ cftfifqui 

tf«4rffli,5i ucro Q^o mdiorcs noucm minoribui junt ^ui: units mdiomm dd «- 
ni4.m mmown coinofatur fifquiodMinAn muH^icibus .ft umusbinih duo^qud^ 
terms ,tr€sftnu aqut j'unttmdtor mvi9ns efl daplus.si wtus dd trtSf duo ad fix: 
f i^^ tnu^r mmoYii cjt tri^Ms.%i wHus dd ^i*4fuor,cr duoddoHo: wtus mdioru 

\- ddu^ii4fHfHinoram (ft qujtdru^lus.si ucrounusddodioyduc dd fidedmimdiar mi- 

I n^m eft oauflus.in fu^^rpAnicnnbus.ft trcs mMcrcs aquifunt q^s ml4on*^ 

^ hi4i,dutf€X deniiiunjs mMordd minorOn (ft fipcrbitenws. si qudtiir m4iprcs 

fivufn pninonbus:nu:or minons eft frfertriqudrtui. ii ucro noutm aqui funt dd 
decern cr liptcfn:md'^r minom eft juf.roaUfdrtuns M in compofttis numtru ft 
muccftMCjiduod^quifitntqudtuorbcrdintldium unius.qufn^dmodufH m 
ditobus coniundu ftf^udtcris.^quonidm quAtuor CT unius dimidium dd duo du- 
fiumjunt cr ftfq^^q^^T^*^'^''^^*'^^^^^'*^^ ** ***^ ^^^'^ ^ duplus tjl dtqucftfquh 
q^tus.Si trcs d^quifuntddqiiinqueb C7 tcrtidm uniuSyUtinduol>HS fefqmtcr 
tiji.fi9nidm quinquc cT tcrtli unius:cont'mcnt tres fcmci, duster tioj CT uudm 
nonAfn.pTQinde unus dicondnebitb unum,duds cit^ tcrti^ CT w"*^ ««^ nMom. 
{ *r ii qudtuorxqu\f^ntddfcxcrq»^rtdmunius'Mtinduobus iun£lis fefqurqudt- 

' tui.quonidmftx cT qu^rtd unius icontintnt qudtuor, dim\dium,cr duimdrnfcx- 
tdfTt cx qudtuor.idco unus mdior continet mmoran foncUcius dimiSum, CT fi^s 
ufum dccwtdtn fcxtdm.u uero odoudUnt deem cr oadudtn unius ut induolms 
fefqid^aduis.quonidm decern cotmntnt Qao:CT*{*t^^f^yO' oCldUd unius dd o- 
iUnQs u>u eft ftxA^eftmd qudrtsAdco unus mdiorum contintt minorem fimAyt* 
ius qudTtdin pdttem,cr cius ftxdgtftmdm qudxtdm.Xtft compofttionis ftrits fro* ^ 
tenfior cudditiut ft duo d itquiptntfex b,me<Uetdti unius cT quarts ut in coniun 
{\iQnetmmfefqudt€Torum.quonum[excontiwtHtduos ter CT dimidium unius 
eft qusrtdfdrs duoTum.O' q*^^^ P^^ ^^^^ ^fl ^^-^^ P**^^ duorumAdcirco «• 
nus dMplexeft b unius infuperqudrtdmdtqueoadudm fdrtembcontinens. tt 
ft trcs ^uifrnt ddfevtemcT nofwn unius :quod in trJjus [cftiuitertiis cQna4m* 
Clis eucnit.quoniamfeftem bis continent trcs yO*^^^^^^^^^ pdttem: cr nond 
unius dd tres eft mteftmdfeptimd.ideo unus md^um continet minoyem bis, cius 
trrn4w/cr fius uiceftmdmftftimdmM ft qustuoraquifuntftptem dodtdnti «- 

nms h9cefttribus qu4trtisO'^frf^^^^'^f^^^'^"^^^-^^^^^^^^^^^^^^^**^. 
fefqmquitrtQrum.quomdmfeftem continent femcl quatuoro- tres qudterndvij 
fdrur^C ^^ ^**^-* ^^^ "^ qudtuorfrnt tres decim^fext^, Z7 decimdftxtd 
uniM tH dd qudtutr snu fexdgeftmd qudrtd^dcirco whus mdior continet unum 



mrnorcm 



mlnarm (imelyC ioiormtem,Cr tres ciecwujjextas dtque undtn Jixdgeftmam 
quartam,siutro c6lo ^uijimtdimiscimjtrts oaauds unius CT unamfexage-- 
fimd»Hjuartdm:utiH tribus ftfqmoHduls coniun6lis,quoniam uHdmmcontineKt 
fmel oSoHdrmfn & oStmarij tra partes:^ tres oCidua unius fimt ad c&o tret 
ftxt^eftm^udrta^tr und ftxagfflmd quartd di cflp cji and quingcntefmd iu0 
dcdmd.bmcfitutunus mMrsmrcontinedt: mnptemjimel^tres dus padUds^tres 
fexagtfimdpjuartds a* undtn quingenteftmdm duodcdmdm. It hec adiccimus 

^uo^s hdrum dudrmn ultimdrtwtmfequentibus fdttntiorjmdnifcfUorqut bd 
fdtur. • 
,j CDuplcx intcruallumtcx doobus maximis fuperparticuUribus, fcf- 
qualtcro atquc fcfquirerno coniungitur. 



I <^ I 4 I 3 I AcdupJcx mccfuailtini. 

I a I b I c I A ctcxqualtcr.b clcxquiccrcius. 



A b duplex inrcfuallum^b c ielqualtcrum, "^ 
^Sint db cint€ruaUmn:a- d quidcm adbcdufUx,&b dd c fefqujter. dicod 
6u^ quod eft ctnpJicem cjfe.quomam cnim a ddbdui^exeftrJoa pefdtffimn^ 



^VXtlQt 



nan continet bu b,igitura ^quutur duobus b.CT quid bferquaitn' eJiddcHzitur 
b conmtt c,^ tius fartcm dm\dum.crgo fcr oaauam hllus duo b ^tdfunt 
.JT /' ^^"' ''^^^'^' -'•'^'''''' ^ ^'' ' ^quitrmtt im dirn m. 



"V- CDuo 



C S'tf^ dbc interudUu,d quidcm pfqualta dd bjb uero pfquitertius ad c, dico a * 

duflu effc ad cqu^oniam cnim ajejquaiur eji ad bier^o fcr odaudhuius duo 4 I 

4equi flint dd trt$ t. Cr rurfus quia bjefquitcrtiusad cii^itur fet eande tres i,^- |! 

quifunt qudtuorccr tres b fofitifunt cequiduobusdduo iffturaaqui junt ad % 

qudtuor ecu qu^cuque uni cidemque <tqudllx Junt: inter Jejint cr *fquaiia. ^ \ 

qudtuor numerus minorum coUe6loru:dufUx (ft dueru mnu ri jciiicet coUeticru f 
rHMTumJgttur ferfraecedentem unus a-Mu^Ux erit ad unum c.gwod frut demo- 

fhdndtmt.potes ey hoc ultimuw etia demo^ari.Qupniam enim duo d <tqui junt I 

qudtuorc'MTJus d fer odduwm froloquium ^quus ejfe duobus c, quorum enun to | 

ta ^ud/unt:CT cormn quoque dimidia/unt aqua, at duo c'Ju[>iifumi4^ius.igis | 

tur cr um4S a^duobus c <equus:duflcxmt c t0iius. quod fuif Jet monftrandum. I 

^ CEx duplici inreruallo atquc fcfqualtcro; triplex naicicur intcruallu. f 

^ I 3 I ^ I Ac triplex intcruaHum. 






M 



CDuo duplicia interuallatquadruplcjc conlung untlnteruallum. ^^ 

"XtTduplcxlntcFuallum. bTcoafi miiicer 4u plcx, 
A c gcminarum duplex inccruallum. 




Csif 4 duplex ad fc.cr t <J«P^'>^ ^i ^'^^^0 H^o""''^ ^ qis^druflcx ejl ad c mim 
auui dduviex cfl ad bjnttir duo b ^«i funt uni a,crrurji,s quid b dupUx cjt ad 
lirttHT Loc^ut/unt umb.vfi d^ c^quifunt wm biquatim c ^ut ^runt 
d«tu> b.ai duo folin^nt<f<imi44ii aiigisur <j q^^aiuor c ^qu^umur uma.at 
quMuor quadrupiumjuntu^ms.'t^tur f^cr dccimamttnusa: quadruj^ejt im* ^ 
Mi cquod not ojUnd<ndum* 

CTcrmmumrepcrirc.ad qucmquotcuaquc volumus.Uccat fupcrpar- j^ 

j ticul arcsatVignare* 

\ 1 '^ j g j 6 I A adctclqu^Ucr, 



"1 ^ 1 c I baaciciquiccrcuis. 



Csit propo/Irwm tcrminum refcrire'^ad quern (ifquxltnum c f^fq^itertium unf 

Uamui Alj\gnarcXafio duo CT friu numrros denommantcs partes fefqtultri 

-^ - dtque fcfquHoni-duofiquidan pfqualtctfim CT frw (ifquUcrmm dcnominant. 

%j duco duo in tru:proucnutqucdu^u lUo c man:f€{iu^m cjl c habac pattern di* 

midiatn fmter cr tcrtiam Aun^o ad cfartcm eius dimidum: CT compojitwj/ir 

d-cr itcrum ad <: mngo j^artem aus tatiam'.crf^^ compofitus b.quomdm cmm a 

c^ntinct c cr ^^^ fAfumdimidum.erpj^erdijfinittonem aadcjefqualtereft.a* 

quomam b contincc co'cvui ^srtcm tcrtiamib ludcm ad c jejquumius c i^itur 

rcpcrtus ejl terminus ad qucm j^m/upcTparncuUreSjriteJuntajp^n^tij CT hac " 

If w 4e quihufUbet effcto^nandiWt.ut fi ^[cnt ^/Ji^^^^jd: jefquioaauusa-fifyi^i 

WfnusdttcotUinnouemo'f^^^ terminus odauAinnonamquc fartem poj]idcs 

ddquem ^ (ifqtdoaauum ^jifquinonum rite ajft^aucru, ^eodemquoquc 

foiUimUo tres,modo quatuorautquoiquot libu<rit ad cundcm terminwm ajfi* 

pubis* . , 

j 640 I 600 I 575 I 54Q I 4^^ 

I a I b I c I d I c 

ytjt ad tandem termifwntpeterentur ajjt^njnjcj^u:cerciui,l€jquiqi^tuiyicf- 
qmquintusj^ fefqui^Clauus ducantur tru.quatuoTy qmnquc CT oao in frinuicc 
Xj" vroucmat eiqui idcirco habet in ft tertum^quartam^quintam ^ o^auam- ad » 
Uw eo efiam tertiamxcompofitufque fit a.vf^am quartam; cr comfofitus ft b, 

Juam 



« 9 

fidmifuMdmieomfofitiu fit e,J!i4m da^tjw cfUudmi O" cmf^fttus f\t i.iko 
d€m9^t4tiQnefuj^oretqi$4yMMi€C6nfkMijunt 4 fr c J fifquitertius, pfqui* 

CSiafefqualtero inceruallo fefquitcrtium dcmptumfuericintcruallii 
Jicrirquodrelinquiturfefquiodauura. ^ 



6 I A cfcfqualtcr.Bcfefquircrtiuy. 



A ad blclquioctauu. 



tertid fdrs ntinor f/?,cr /^^ «» J<< frpximc muior^O'fi duo in tria daxcris (unt fix 

kcirc 



Csit c terminus ad que per f>i4€cc4entem djjignati fmt dpfqualter: ^ bfifqui ; 

tertius Ab d cfifqualUTo fuhiuco interuAUt^m pfquitertmm b c,rf lifto interuaU ^ 

l&d ad b:quod^dUoe(fe jijquio^duum.quimidm enimaeius quod ejlc eft fifquaU '' 

ter:d fimd hdbctc ^ ems dimididm fdrtenLquare per odaaamhuius: duo d *e« * 

qui (Unt tribus c.^ qudtuor aifex f ,^7* odo d dd duodecim c.Rurfus quoniam b hu U 

ii4S quod eft c fifquitertius eftib igitur hdbet infe c^ cius tertiam partem. quo fit rj 

ut per Cdndem ottdua-^tres b aqui ftnt quatuor c,^ fix b:oSio c.^ nouem b:ad t 

duodecim cModo a quidem aqui erdnt dd duodedm c:i^turo£io a ttqulfunt dd '; 
noue^ h.fer dedmam igitur d contmetl ^ cius o^duani pdrtem.eftque d ftfqui . | 

o£iduusad b ^ frofofitu^, p 

2^ Clfiteruttlli quarta 3c duodecima: vnam eius tertiaru redituunt. 

C Sdm qudtuw interudBi quart^ttotum coflent interudHwm. ^T duodedm eiuf* | 

dem interudiU duodedtmeitotrnt itUem complcnt inttrudUu^quare quatuor quax V 

t^ CiT duodedm duodcCMueiddinuican dequdnturAua iptur qudrt^fex duoded i 

ms^^umd qudrtd tribu^duodedmis junt deque.quorwm enim totd aqua funtz ^ 

^ coium dimid\a.Kt quatuor duodedmoiduodedm duodtdmarwm funt u/na ter- ^ 

t\d,quaxe0' una totius tertu:cui quidem duodedm Hie duodedm^e cognofct0itur | 

t^e ideyn 4tque aeque.wnd \yXur totius quarto, qua tribus duodtdmis aqua effe | 

moTi^dta eftyO' uma duodcdma iUi adiu/H6ld:totius tertiam partem reJHtxtf^f.E- I 

quant ur enim una quarta CT una duodecima quatuor duodedmis,quod erat mS^ | 

ftrandutn.U qudmuis hoc demonftratio /equentis grdtia particulariter fa(\a fit: i 

pojfis t amen modo con fimili monftrare omncm minorem cuiufcunquc totius inter^ I: 

Holli fartem^p^oxime mdiorem pdrtem €ffic€re^i adie6io una partimn a numero i 
qui ex du£lu denofiiinatiom^nutriufque par tiumendfcitur^denominataM totius 

iior^O*f^d^ 



r 



iccifco ufu MU ^ut9n4/ixU,nn4mrt]btmmificumUf fi^ 

crhoc paatinftqufntibui utfubkaa mpnflrft^U, 
tcftidyjixu unmfian^mtffkiunt. 

Qjiiar^ cr «ic'i5»w • uTurnqBortam. 

uxta CT triccfimd «»^«» qumtam. 

€Du;t proportiocs fcfquioclaux^minorcs fut fcfquitcrtio intcruallo ty 

-j — 5-; j — — — j Z^ j [ A c duofcfquioCtaui coumocti. 

I ' — a b 1 c 1" I d I A d icTquiccrniiminrerualifi._ 

CCofttinud ferpxtam qudHidruhmctices duds fifquioeidddas inter d^- 1: hnut 
aftfquioaM4Uiht ddb^ bfrJ^^uioaMHsdd cf^ ut d ad d^ify^ittftum mm 
uMm St0ddd c minus effc mtcrudllumiqudmdddd. Q^otMm if^f$ dfcpjui^ 
' ^U4US fft dd bicr^o per o^Midtn hum odo d ^tquifiat dd houembjid o ^«^ 
4 betidm CcCquioaM^ids i(i dd C'^nandm o^lo b tdntuffffrntdtquc nouc c. U 
cum unui b Muyiftt.unic a oadu^aus-.n^o nou€m b ^quffunt dtctm c CT 
cddUa unius c.Ktqui Hfocw b mtrnfirdtifimt ^uicffcoao 4: ijtrfcf ofl04 ^qui 
funt decern c cr oddUi^umusM dtcan CT 0^4^ umy,conhM€nt oaofcmd^ 
lus qudmm cr f'tus undm fexd^efintdmqudiatm.eiid fer dedmdm huiununusa , 
cofttinct umm r, undm nw> quansmy cr 4www [ex^iefimdmqudrtM. cr »>m 
qudftd cr undfcxjgeftmdqudrm:f>er frrfcciufcm mmuifrnt uud fcrlU* xowf 
flaa cnim qudrtd cr duod/tanuiunAm xatuan.duo i^UurjlfqmQadui miMUs fut 
una Mqmttrno interudUo ,quod Odf piOMJhdndt^fn, 

CTr«fcfquioaaui:amplm$funt lci"quitcrcio,miDUJ autcra fcfquaU 
rcroinccruallo. 



I 7^9 



5 1 1 j A d trcs refquiodlaui coniucli* 



I d 1 



Cs'mtdbcd tris coniun{lifefqmo6lddHjHtd4db frimtu fitfe[qmad$aisj^ dd 
€ fccMtdus , cr c 4d J tntius: Sco prim# dddc dfUfbus effefepjwtatw. Qj<ow 
a emm dddbcrbddc duo fint fefqmc£idM:fr^0 per oSldua biius ^od aqui 
fiiHt icce c cr ^^<<<*^ unius.SedQ-cuwc iteruad d^fitusfttfefqui^ajafusier^ 



fcfCdndcoCip e gtqulCuni noittmj^^ ncui c ^qul itci i Cr •fl^^f unhs^cr 

dcci Ciequi undecim d(^iMlmfcit4uds,^ cuc&o cSm^c c contlneant noucm 

duri0per decimaJbrntiSy^na pRdua c,cctm€t cClaaa d ^rius eRdua fartcm, 

h^ tfi undm fcxagtfimamqu^irtam* due ergo c ^ una oCi<uia mausi ^quantur 

u^dccitn dytribus oiiaids O" uni ^xag^m^equ^irta.O' f^ decinutmut umdcdm I 

tm oilaua Q- toufcxd^cftmaquartdtMd oHoiimadd d'ftd umdecm cotlModo I 

nos fcmcl^tres tcru fdrtes^es oSU.C uHamfexagejimMAquar. unius^tres o J 

£lonorufdrtcs:4mplwsptUTtute^ufdrtt.SMj^crantetti tns o6louarij fortes: j 

tmuun eiufie parti triete t0tiujjboc cji tertu unius parte. a fortiori igitur tres o~ ^^ 

aonoru partes ,er tres ^cim^ smuSjCr t(^^ ^4. i^ amfUus fit tertia oilono. ^ 

parte^cotinet ergo ii,tres o&4u^un\us 0- uma 64;^^% fitntlcT ^mplius eorupdr 

tc tertu. quxre CT ^ continet d CT 4mf\ius tertia eius parte, r(l ita<p a ad d 

awpUus Icjijuitertio mtiruallo.secudo dicoa ad d minarc ejje fefqualtero interual ! 

lo^i^am n cotinet S CT* 5 o^auastdeefi ergowta oSaaa adcSpUdas 4 oRauas^ 

qiftit o{lonciu dimidiu, At uero q; fuperant:} oClau^ u/nUis, o* una ^4. minus 

ejftciut dimidio wtius o{i.U4ie.quure CT ww*/fo minus cfficiit 1 e^laua, 1 1 ergOy CT 

3 oddu^t^^ una64.t^vii4i:cStinet. Bftmel cr minus o6lonoru dimidio^trgo per > 

dccimam huiusia cominet dfemcl ^ minus eius dimidio,eJl itaque interualliwi 

a add minus fcfqualtero interuaUo* ;r 

'^ CQuatuor fcfquiodaui cotiiii£li:rerquaIcerum fuprrant inceruallum. 

I 6s6i^ I I I I I 4Q5>6 I A c quac uor Iclqmodtaui 

I a |b |c|d| e \ coniunai. ; 

Csintd bcde 4. cSiuC^ifejquio^Md^aad b primus ji:^ ode feciduSycad d terti- I 

us,^ ddde quartusidico ijuomam interudUu a e,dmplius cJlfefqu4dtero interudl ^ 

io.Nowi utin pr^edentiuisu eft:S d JtqUifimt it dytribus odauisy^ kni6^u* l^ 

nius.O' * * <i>3 oddu^t0nus ^tt^na 6^^i»iiidnturix e,6 o^auis ,quatuor fe~ l^ 

xagejimjfquartiSy^umi quingetefimce duodecim^e.ergo 8 aiicquifut ize.6 oda | 

ui:^y4^fiXuge funis quartisyCT ^^ quingentefinue duodecim/e.At uero iz 6 oda ^ 

u^yquAtuorfcxagejim^qudrt^ey^ undquirtgentejimd duodedmdi^cotinent $ /f- * 

ntd cr ampuus q.odonoru dimdium.quomam c^tinent odofemel CT dimidtmn: jj 

^ wjuperjix odduaSyquatuorfexageftmafquartdSi 0* imam qmngentefimdm i 
duodecimaml^iusAfftur perdedmam huiusid cStinetefemel cr dmplius ' qudm 
ems dunidium.frpePdt itdque d e interuallum quatuor fcfquicdiUns coniundMm: 

fefqualterHTH mtenMUuit^quod crat monflrahdu^y \ 

Ctf. Quin- •■ 



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M 

C-^iiloqucconiundirefquioctaui.minurduplicimteruallocoiunjfic.io 

q«i»(fJd«n tr^-i prwif/rftwoa<i«J:prrdcci«4oa<*«4bi««5 minwi fuf /?/^«4ittr» 
wfaj.4iio.cr <!»«/?'!««" rtfquioaaM,(f4iiuna cumtribus frioribus quimque ca 
flcnt:feTdecimamliftim4mhumJ,minuiCuntp[quitertio.coniunai x^tur ^hw. 

quertfqmocidUi:mmuifuntiiiflKiinUTiitUa. , ',. ■ ■ ., 

Cl-cjcrrornrn"n--f-'-T-'"^'"''"'^'-fiO'"tvnodupIicimccruaUo. 

- rTtT^^TTn~l ~in~ l ^lil44 I A giexjtquioctau. „ 

i-^f j — b I c L__a ^ c , i^n^ g jZ*«iLH5^i. 

«i.,tu f mcruallu fcx fo«mn7f7r7r^^«<ort4«cru,it4 «r d *! t ;if pnm«i Jcfqui 
ocUu.f,b ad cfuuius,c ^dd tmwi.dad e'j«^rf«.,e4d;<iw«t«i,f ^trm jdg 
ftxfui-.auo <iMo«M J <d e,cmpli«x c/l iu^\mimrH<Mo<iupmam<i^bmui 
\>cUui,it(i.aho ^aoaauihuiuso(\oa^qui^utnoueh.o-nouc b-.yir e^dtm 
s^iuu'ut due f,cr oa^a^ unius-v decicG" o^aua unius-.^qy^fut und:ctm d, 
n<Ui oa^ui, cr «"' /fx4er/ir«<«.j«-«rf^ unius CT undiam d,tus oa^u^,crUn* 
U ««iiw4fl«4rr4 uniui-.xqUnm duoitdm e.fix i^auu, quatuor ftx^S'}"*")- 
quJtiia-uni qumzt>tt(fim^duodedm^Modccm <iutet,fixca<tu<e, quatv^r 
Lai,ffim^quiH^,cr unjquin^tntefim^^duodccima-.fquatHr m<i«.m;,drdfm 
- oa.U4L d«e/fx4?w;.mf<i«4tTb,cj«mq; qui7>qujStf,mifdu„dccimis,<y^ urn qu^ita 
miW.fmtncnag.imafcxt^.M uno dcdoiUu^.unu ccrinctintcsru.crtnjUpor 
duJoi}Mt^.quoht itcrh utduodcdmx,}tx oaau^,quatuor fexas,eltm^qujirt^ 
O-MW quinmcftmd duodccima-^quituT quatuordcii},dujbus oiiauu ,dcceJtXA 
g,d]m,iquani>,quir,qi qutngct<ftmifduodecimi>,o- uniqmrt<e miUepm* >.W4|f/t 
,,U(xt^.At ucro qu4tu»rdtclf,duxoaMi^,duefcxj$cfm^U4Tta.quiqiqm-^ . 
P«/,l.«rfJuod«im^,cr"w l^^rta milUfimA noM0imu[€xta:^qua)ut qutdca 
% J- odo oaMds dus,\^c efi ^qua{iitfedeci$4uodeafexasefmtfqumis,quide 
am q,m,gcttftmjduodccimis,fcxquartis pidlejimis »onas(ftmufixtis,a-''Mm^ 
(effimjtfeci*>id<e millejmafqtigentejm^ftxagefmx oaM4<e.Atfidccm v 4«» 
decimftx4i(fu»xquartx,Cr rdiq, fequites fafticuU-.cotinetoiioks cramflt* 
us mturf^rdctimam huius 4 ajJ^ mMus efi duflici inttruallo.lex^giturftlqmo- 
i\.twmMousfutuuodufhci VitcruMo:ut mttdtt ffofttio.Ufi hxc i^pojitwquo 
adluadan^atUnmninuUiijMifficilieruideatur.hocuareo futniet <p tUts 
fromphtudoyutrndique hMiUs oHmx r.oiuccr i(dm<e hutut dfmf. iccmo 
tMf cSfetMCumque ait eosqui in mufcis micuUtionibus CT "»'«^ contci^Ut'.om 
bus ft exerdtart uoltnf.multos frfqu^lteros ,dtw^t fefauitentos,^- "Uos fiqucv' 
iteijuf>rf4rtuuUTes coU,gere:quou)que ufus ccUigmdjrtm fro^ortwrnm ifjii 
am fit fadus {eru!us,^atfnfque cr qwf '-"« irf'^'domtfiicus. tnin 




I n 

CPrimi clemcntorum muficalium finis. 

Onfonantia ; eft font grauis > acutique mixtura fuauicur^ 
vnifotniiterquc auribus incidcns, ex multiplici aut fuper^ 
parriculari rariclic profc^a. Diffonantia; eft duonim fof 
norum non fe nacura fuauiter mifcentium 3 ad aurem pen 

^ t ueniensafpera,iniucundaqucpercuf$io. Tonus:eftconio- 

nanrix priDcipium,cx loniad Ibnum IcfquiodVaua proportioneproue« 

niens. Semironicm minus^ quod & dicfis dicitur:cft coni pars, qua fef- 

quitertia proporti.o duobus tonismaior eft, Semitonium maius>quam 

& vocanr Apothomcn :cft tonircliquapars , & qua ipfc Icmiiomum 

minus fupcracCommarcft quo fcfquiodtaua proportio^duobus Icmito- 

nils minonbu J maior eft, quod & idem eft : quo Apothome , Icmiro* 

nium minus vincitac fuperar. Schifmareftcommans dimidium. Dia* 

chifmarcft dimidium femitonij minoris.Hemifperium muficum eft in* 

ftrumcnrum, per quod aut neruo aut chorde vt dccetfuppofirum/e- 

mitonia,tonos,con(onantiifque5& confonantiarum particuias^ ad fcn- 

lum perucftigamus.Sonus emmelis is eft;quo apcc vtimur in melo.Ec* 

melis veto is diciturrquem meIos,c©ncentufque non admit tit. Equales -J 

foni ^tquc fimiies dicuntur ; qui ex eadem inrerualli proportionc na- ' [} 

fcuntur.Numerorum atquc interuallorum pars ea maior cft.quac a mi* >^ 

norc numero denominarur.fic minorrqua: dcnominarur a maiore.MuL 1 

riplcx proportio maior eftrquam maior denominat numcrus. & minor; .1 

quar dcnominatur a minore.SupcrparticuIaris proportio maior cftrquje 3 

a maiorc denominarur patte.mmor autciqua: dcnominatur a minore. j 



I CTonum fuperdatamchordamcollocarc. 



b| 



C5if a he chorda quecunque fuj[>ra quam iubeaniur tomitn coUocarc, diuido fer H 

tntidtn fetitionem chordam a b^in noucm aquas fortiones :im utcb illarum no^ ^ 

uenariwi odo tcneatyCT a cunum, dico quoniam a b tTcb y tonum cpntmnti I 

conjlitutumque fuf>rd damm chordam effe tomim.Naw totum chordcfpacitwt a b: ;:j 

continet fpdciu^ c b tT* in/upcroAtuam cius fart(}ftyqucd a c uni ilLruntoda- J 

U4ri4^ <eqHafit.i^it^r fcrdiffinitionemfffacium u b:€fo^dount,fefquio{iauiitnqu€ i ■ 

cjl fpacio c b.quarc |>cr j>n?n4w pctitioncwica tritfim totiui chcrdAca b adfi- '>, 

I' 

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num c b prorwt/o. ij} itnque tonus in chofdd d fc^wi intfo^doJ^efqtdcihtHdiiu^ 
utione conftfiiticoUcattus^ 
CTonum tono,fic quotquot libucrittin data chorda {ubiungcrc. x 

U ct dl c l— 1~|~| 1 I 1(1'^^ 

Csitddd doiiddbi m<jud frofcfttumfittr^s ccnfequcntcslcncs fkbiwtmt. 
vd rftor prr urtiam pftfticwrmCttt m fr^tctdcmi faC\wn efl)ffddt4m Wtivs cbor* 
drd b in noucvt aquds fcrticncs . cr ^w ^<^^ o6hu^ fortictri^ fon^^m ut t c,\ 
o^uds iUdfum nouem fdrtium tcnejt tndmfefiwt efl f rr fnactdtmim: dbtT 
c b effc wnum . O" f^ €dfii(m fetiticmm : fdrtiorffddunt cbm nouem ttquds 
PortiQnts.cr w tcrmino ethu^ j^articul^ powo i:im utdb contincdt edo tdjum 
^^JrtMm qudrum c b ncum contintt.f^ ^^ceiintm cbdid b/otMt Wn$m, ifl^ 
queidmunitvmo.wnus utius fubi$0${lus. Kurfum fpddum d b conprmli modo j» 
nouan ^Uds fmicncs diduco^CT ««ftiw odnua ft&ionis littCYd e defigno: ia Ut 
€ b o^# tdTum fortiim ccntincdiyqudruw dbccnt'trM ncuint, fnfr^ccdattcnif 
dbaic bnfonatwmm. fum \ptur\n data chorddd btres continue /LttWKfii m* 
nirfdlicnd bcb,cbdb,db c b.quod crdt vrofofttmn.a' ^oc fotto auotquot lubtt 
fuiisinprc : quitn fddUimmH ep.itfiid fhifu txferiri, defrchatdtrequc at f ids, 
' po/? v3/um tutus chcrde d b ftp f one hcmfphmuw chorde a b infpto Cfitn irt/8* 
L perprtpat rrfinitque pdrtu:uU c b:crfinfis iudido deprehcndcs font tvtius db 
dJfonmm c bcUt toni intcrudUufn.quldfxhemifyhmmt trdmfers dd mtnm d: tx 
fdfu chcri^ Umm txinum deprehcnddsfrd ex totius d bfono dd fonum f drti- 
c»U d b duos tonosy duoTumque tcmorum intnudUum perpendet duditus. CT hoc 
fdetofhipcum iudidis quotquot uoles ronos dcprchendenJos committcresrcr torum 
mixtuTds turn fudues ,tum incondnnds(quds duditus tanqudm offenfits hcrrct it- 
fugttqiu)deamendds, 

CTonoram conrinaatorumrminimoj numcroj afsignare. 3 



f9049l5H-^^l46656|4t4'73^l3^^64 | 5^76o| Qu inqi Co. admuic^ 
^J I o I p I q I r 1^ I contin. minimi nu. 



p I q ._ 

6^ 61 I T^l^l 5»^4 I 4^'^^ I 4Qp<^l |Quurorro.adiouic£ 

1 I m f I c onnn. minimi nur 

Trill rono.adiDuicic 



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•71.9 I 648 I $76 ( $lt 






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(conrin ^mimnii nu* 
f Duoru ro.adinuice^ 



concra.mmimi nu. 



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n It 

^sliuQ^tn$f^udtumf^hu{uedut quotlibettonifintcetitviukin mruc cey^ituti: 
ficewm minimis numeros refctientus. Ci»#wx4W cnim noutm cr ^Qo mnmi 
fimt numeri tomiiuco noui'in p CT proumat a^cr »^«i w ^£io cr ffouenua fc, 
er ^*«« /? cr ^urnwt cf^n uic^matntia tntij CT frrfextam quani4rithme* 
tktfintird b CT't cfimtduofcfquwihtuiin minims numtris conUm&i:^ fro* 
mieiu0tDm mminmts numnis conmuatixt finouaniucoind b € ^ o^ioinc 
Crftri<mti t fgi pfT candim intit defg trafimt in tninifmis numetis fifqul- 
cfktm c^nftitmlquare d e fgfunx trium ccntinuatorum toncrum mntim numu 
ri.U ft ducifnoutm indefg cr o6lotng C" Ji^rg^tbiklmiftridiqH^frius^ 
hi)s.imquatuorcom'tnuorumftfquio£hu<nutmnm\(unt7mmm.quayc^qu4i^ 
tticr continuorum tonoruw.ft fi rurfum hocfaeioducis noum r» bi k Im cr ^ 
&otnmcr fing^nt n of^qr si'iyft trum quinque continuorum tonorum minirtA 
msvteri^cr bo** niodo quotquot tonorum uoltsiminm^s numeros rcftrias. Att^ 
men in modis mufuu totco7thMefuhm/ng(rtofus non efl:-edtoms fimttoniafubf 
b$itpp$tur.dcquibus foperiorpo^nrius^uccommodufqueexfiaandus cjifermo. 

4 CSpacio quotlibct per quotlibcr rqua fpacia diuifortoriuj ad coram 
proximi (cftionis partem minor eft proportio , quam eiufdcm partis 
ad coram rcliquam proximo fcftionis partem. Quo fie vcquanco tonus 
tono fubiungitur acutior: tanco ipfum contraftiora concincant fpacia. 



I A cl d] 7} f| g\ h'f i^ k( bf 



^choTd*im^eruum,tibldmy ejfUtum acra^s tinnulum cr quidquidfinum eitt in 
harmonkis modis(ut iam quoquedUlum ejl)ff actum nt^cufamus: m quibus 4- 
»4iDgf<r ratio wmynaturdrnque fituat edndcm,sitergoa b totum ffaciumfera c^ 
c d,df,f fjCr reliquds riOUimaquasfdrtesCutfitihinteruallo tonihabendo)liui* 
ftm-.quotquot enim alids fofuerU idtm uJuerit, dice nunorem effe propottioncm 
abad c I quam cbadd b.Uam cum a b fofitumjlt in noutm d:quas fortes per 
medias notns cd ef^rdiqua^diu^m :c bcontinet plum earum fartium otia: 
quaru abcotbtctnouLergoabcotinetfpaciucby^tius odnud f arte, fid O'fji 
c b 0^0 fartes parti c d ^quas cotfncatiergo d b cotinet fartiii totnrumjolufiptL 
ergocbcSdnetfpaciiidbyCri^fupereiusfeptimdpartlatoihua pars: per dtffini 
tiontm nunor tji parte fiftvma.igitur iteru per dtfjjiwtioned badcb proportio fun 
ferfortkularihminor ejl^cb add b utpote qu^t a minore parte denotninetur. efi 
emm \hec afepttmdfartefefqutfeftma ^portio:iUduerodbo^udfcfquio£hud. 
tthancnon mod^ infuperparticularibus : uerum cr in qutbuflibet medittdtibus 

C iiij demonjlrat 



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M 

Por«o«*««f«« cp»/piofU«r.c<»mLcn«w .wjo bwc «x«fc*r«r^ »«« 
prr <i«4rMm prtifwwcw dJ foam f^nmsrdumfoH4t:p<nt^tmadfmm Mu 

quonum rpjcwm cbim noui ^lus pjtrns p4rHrttitT:quirum (juelibtt mam ait 
}pacv> c Iqmi dus oOtu^ ifinM q«f!.t*t tanm ejl noru^Anuwn nmun 

dicatittm.mxmfcjiumtH iptur ut qu4H:v anas ano fubii*njUur Mgttor: mna, 
ipfum contTddmU contintant fpdcia. 

CMedio exrrcrait«um toni fpacioin duo xqua diuifo : t^us minimi S 
in duo («catur z<}uaha. 

f| g|_I^l Ll__»iL__N 



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CHifc cr /Ja««wfi4 btfc%»«>frirr »t ewittJ »««fMe in duo ^qud rdttone gcome^ 
tricadtrmitiirznoHduttm arithnetkoL rum a-^ f^""^ tom({ut mtcruaiium: xn 
iuo ^qus Tdtiome arUh.dxTcmptii efl.dmido ergo /p4au d h utjrius.m noue^quA 
fpacu:fernoaisaCydeyfg,hh\^bym4mfeflum cjiabv cbejfctoni cxtrcnutdtcs, 
tonunMuec^tinere. iic9 crio quonum domdio hxrum extrcmitntum intcrftitiOy 
in dm ^qt$d f^er fignum in ttrcm}^to:tonus minme in di40 ^|»i fsrtidtur^qttoi* 
qu^ fonusdbcrl b^qtidlis non fitl b, CTcbJiuidQ emm jinsuldreUquorum 
c^o ^qudUum fpdciorumcmfimili modofer ^qualuy^crnomi wi,«,<>,p,^,r,y,t; 
msnifijium ejl Mum ffdcium d b diuifum c[fe in 1 8 ^qu^lidfpd^id.qi^ funt a, 
l,c,m,dj^c,Oy CTTfl^f*^' ^^ ?^ j^^ectdcnccm minor efl fropottio dbcrl bi 
qukml bcTcb.cflfnmh^ecttfquifexmdecmi: iEdunolefquifcftimdecmi. 
yjocfl CTjd tonus hocfdOo bi duo^quddiuifus.cr fi'ndblbj^lb cb perdiffini 
thnem ddinuican in^qudlcs: quiquidefiniccmeksfunty muftcoquemdo pfrp4* 
ram aftu 

CTonl fpaciohocmodo diuifo,torius medic fcftionisfonusrmaiori? 6 
cxrrcfBitonifonura acumiac,minoris vcro jrauitjtc fupcrar. 

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Csit 4 t «t iw fraccdenti Urn diilum eflydiui/um : imut dhadcb rccrefct to* 
nwn*dico quonutm finus I b acufMncjuferat (bnum a t, quodque idem fbnus I b 
grduimtc uincit c 6. Sam a b totum cftcrlb dus fjcrsiergo fer qiiartam fctitio- 
nem a bgrduiorem fonum edit CT I bgrdcUiorem, CT f^ eandem qumum J 6 to- 
tum eft dd c b: finus I b grduior eftjono c b.fupetdt ergo I bfindius toni extremum 
actwtineicf minus grduimte*quodfr0fofiti0n erat demonjhdndtwt. 

^ CTonum in duoxqu3,ccrto,conftitut6quc numcro, diuidi: impofti- 
bilc eft. 



^ 



CNdf fer $ frim huxus^nuMfufeTfarticuldTexnteruJBM/influrdiiequdlxd dm^ 
mitur:CTf^ i fetitione qu^e froportio fpdcion^ninterudi inter pyeaquoqueejl 
f^X fini ddfinum. Atqui tonus ex fuperfdrticuUri nafcitur exinterudllo nafatur 
enint ex epogdod yfefquioibfuiquerdtione. igitur tonus minhnein duo aqualia 
ditimitUTydiuiditurque, imo uero neque in plura duobus aqudiut in tria autquds 
tuoT, EX quofdcUecognofcitur Arijioxenus muftcus dumm iudicio cum^bt com* 
mlttensy ferparwm effe frobdndus, quifimtonidpcus quiim rythdgotici,non ar^ 
b^dtuT effe difniJUo tono contra^iiordifed quemddmodum Jemitonid dicuntur, ita 
quoque cr e(fe integrd tonomm dj^miiw-Ncc minus M^prtianus felixlftmili Idffus | 

errore dep^eheditur:qui non modo tonum m duo ^qudidifed cr in tridjO* >" ^f^' 'l 

tuoY dirmit atque patt. seatt enint mprmis tonum in duo aequdlidiqu^ iccirco • 

hemitonid uoatt. sccimdo in tridi tT e^vrum terttarum qudmlibety diefim tritemo-' 
ridm numupdt. Tertio in qudtuonO' hdnc toni partem qudartumjuocat diefim te* , | 

t:irtemoriam.quod fkc ^efes numc tcrtue^ ntmc quartdetoni flint partes. E/f enm ^ 

tritos tertiusytetartos quartus,^ meros parsfiue morion pdrticuU* vonit cr '^* *^ 

tidm iiepos acceptionem :ut eaipja toni iertia CT tertue dimdia pars dicatur.^t ^ 

rur^um prmi modi prnntque dcctptionis dieps whromatiats uocsUypcumdds uocat i 

endrmoniatSytertids uero endrmonici diuifioms hemiolidSypartim Ariftoxenofimif p 

liter ypdrtim dutem difftTwliter.snmliter quidem:quod Arijioxenus torn dimidlum J 

pmitomum pondtyO* toni terAim,dieftm chromdtiaim uocetfed chromdtis moUiSy | 

0- toni qudrtnmydkpm endTmoniatm,dtdiffimiliter:quod toni qudrtzim cum pro* * 

prld quarts m^^etatCyUocet diefim chromdtis bcmioli j. Et cate uel Martid^us in '] 

tertid diepos dcceptione nouo errore Idpfus putnndus eft : uel twndem putnffe be- . :; 

tffitoniumc dieftm endrmonici heiTnolij idim effe. ndm cum omne totun^tri* ;i 

D bus '^ 



\i 



■% 



M 

hus fids Urtii$ inttrrctur : unm er^e tcftijrum fir terti^ mdieteUem Mius <fi- 
mdmm imfUreneceiji cfl.Scd cum h^ /e fitis fdfa tfft prodsnt c nuaafuivar^ 
U c0h^iant:non (flcufin his dMiui ftrmdftt frotndfendus, sic enim qui ftoli-^ 
dim finfis iudicu0nfeiii$attts,\mclUilim Tclb9<iuwn:f4cUt ex difd^lmnrwm d- 
dytisfe exfUfis fintimft. 

CQuicunquc rmmcnis in tcrminos toni ducatur: inrcruallum toairc* $ 
bnqucr. 

€NimcHm tonus crtoniintcruaUumin propcnrtTontf^pjuio^uacoiiM^:Jidtcr '' 
Tfuni crunt tioufm4icoaoyai4tquicu^qtlealtj <p4i]i7muftbifropcruJnc reffon^ 
dtant.ktcim ffT fiftftnamficumdiaTithmtticcSyft unusidemquenwmnus duos 
nudtiflictt.froduaorum ZT multi^liaiwuim tddtmfit po^oriio: cr^o quici^tqut 
numaui ducctur intmnmos fc^quio^u^ty [cfquioihtudm produc€{^jciinquitque 
tmmn stqut toni intcTUdllnm.quod ejl papojif(*w. Et nw modo dc fefquio^uo 
Cr toni intnuallo U ffntxenium ejl:fcd v ^^ q^^olibct altcro intnudlo. 

CO mnij numcrui.cxrrcmorum toni differentia conftitui potcft. 9 

Csi cfiim tjmus in minmis conjiituatur utfunt nouem atqut oflo: monas differ 

Jr rattid tj\pm noucuixnus atquc o^andrius fiU unitate dijfcntiant, it ft ducatur 

' viiMTius in noucnariwm cr o^^ndfumyCr froduamtur d^r^'-t^ ^^ccdcntcm 

dCT^ ^Tumt toni extremd, Ktutr^ fcr nondmfrtm anthmctice^, quod fit ex ti* 

ndfio in nouendmm : tantum ejl qudntum quod fit ex binario in oaonanum cr 

t$nitatmfcd bindrius in unit;atcm^per commune f 7 oloquimn fit f/umprodudt.igi- 

l«r cxtTcmomm toni d cr btbrndiius diffacntid confiituitur, tt ft ducatur ttrndi 

rius in nouem CT oClo:eodem quoque drguwento ttrndrius extrcmorum toni diffe^ 

fcntid conJlituctur.Uitn quicwnque dlter mmerus in eofdem mnmos tvni ttrmi 

ncs duceturudem extremorwm toni differcntid c^nftituetur.lldmfejlum icaquitjl 

omnem n4dmarum,extnmorum toni diffrnntidmconjhtuifoffe.tt qudmquam ita 

e[i : fldcuit tamen ?hiloldo rythagorico frintordium toni^frmordidlemque eius 

differentidm teYnariumc3ftitune,qui fttmui cubum d frmo mpari numero pro- 

aeAtydc gifwit . quod is numerus dfud rythdgoricos mdxmehonordbilis fuerit. 

fiam cum temdrium ^iynumqutdem imparem numerum tertio duxnis : noucm 

confurgWHt.qu^ ter4uih:ftftimfiiprduiffntiffdlket cubum a fnrimo impari red 

i$0tt>iti'j dd 14 tono di^ant^tvniqueclduduntinterudllum:^ horum differen- 

tid trntdrius conjhtuitur. ejl atim trmsrius (ummt X4 p4n o£laud: qua eidem 

ddie^hi [umme frimum d temdrio cubum rurfus xnjidurdt dtque ferpcit, U idem 

vbiloldus fummdjnpj^evt CT u^nti in duds fartef (ecuit 1 3 cr 1 4* qt^drum b-rc 

tnulcri 



n 14 

mMletilU u^o minor hdhftuf.trTi^tncp^^dmmAfchmes fddt^^^ efitoni 

f<nrtio mdoriilUm Mtmfddt fmdfium Hcpos^qiue ejl4xm iecifi^ miner atqyc J 

forthg^ quam ffflmus fi/mttmUm minus mmcupMmus.tr Ulatum iu4rum \ 

fdrmm dfffcrentuun ^ute ejl umtnsifddt cmmattspindfUwuscd iehisJitU. ] 

10 CNumeridati partem qaotancanqucrcpcrirc. Quo fit vt cuiuflibct 3 

numcri pars ab ipfo fie nuincrata: ^ a dcnominante ciufdem parti? fit J 

dcnominata. J 



I 27^ \Tn I 1 306 I i8 I I .136 I 17 I 

I <= ^ I I s I ^ I I ni I "i r~i 

t 17 I 16 I I i8 I i-y— [— f ■ 17 I s } 

< ^ I b I j c I f t f k 1 -r— 1 

CQ^tf «u enim uumtrorunamerSttuqul exeflarcs in dnim^funt, umms infcmbu 
lis txifldtimimcrcru tamen numndtoruqudlts in fhyftcisfint motuu numeric in 
geometricislinfdrucpl^^oruyinaftrolopds ttforu^in muftcisdute tonoru dtq; 
intcfudlloru unitas dut umus unu feCtionindfitSit ergo d quicuq;numerus cu^ 
ius una fortiudenomtndta dbb qu^fitn fit:refoluod in^fuds fortes denomina 
ms db b^ducedofcikit dtncmindtcmnumtn d in denominate fartis b,im utpre* 
ueMdtc.m4mifdluefl^f dries c fitmd dencminams ab bi^qudri toti numero djlui^ 
do iptUY cperbC* ftouenidt diditpi^ffe parti fetitdyCT numeratam db aMc 
fjl d nuero denominate d.<:iuid enim fit pars a denomindtn ab b qud petitd etdti 
patet^quia a bid perfeptima petitione reprodudt cquare d:efi pdrsc denominam 
ab b.dt c ^qudturAptur ^ d pars ejl a denomindtn db b^Sed c^d nueretur A 
dipatet.quid dinb produdt ciergo perfextdm petitione c diuifo per b^redibit a. fed 
O* ^ode c diuifi pcrbiprius ueynebdtd.nuetatigitur aifemeld. O'fic^pofiHo atq; 
fuHCorrcldriummotafunt.ttquauis hsecut mdjhetur per fadlisfitiufus tamedus 
dliqudtolatetior^ueexeplarisdedu^ofddlhcuiqir€ddetiUuJlriore,utquafitnfit 
uerbiedupifummedecemc'fcp^yp^sfhctndedmdiducoutriufq; denominates in 
fdnuice^ocejl 17 CT iSi&fi^'^^Jt.fextiededma.tot enim fcxtafdedmds con 
tinet nuerus 17 refblutusAimdo ergo x'jxperi6 O* fuenient 17 fextadecimie: 
qujecrunttotiusfummepdrtium ^jzparsfextadedma. quarecr^^^^ H T^^s 
itidem fextadecimd. at 1 7 fcxt^ededm^ unitatem continent ^T unam fextam-^ 
dedmdm. efl itaque unitas ^ una fexmdedma ; dati numcri 17 pars fextadeci" 
mdMfinumeri decern c^^^ quareretur pars feptimadedma , duco decern CT 
0^0 in decern ^feptcm CT produ£ium diuido per ly ZTuetdent i8 feptimdede-. 
dm^^qu^ unum complent integrum,unam feptimam dedmam, tfi ergo unims cr 
und umtntis fcptimdiecimd: dMefimmi decan cr ocio pars und feptimddcdma. 

D ij si uero 



J 



siucrcfimmeiecem trhtmuol$ fortmcOgUM fefmreiiHC4iiinoao.& 
mu94 oToiuaumfisintJmdo fa ^0:0' ^^i^ew^nt 17 caau4e,<iuaiuAJ unu 

Icat txmmH. bolrnt tmcn qua fwticwUn, exmflarique^ dcmvnjtraionc fynt 
QftcnpuddfiqucntU fonius m momentum. 

CSiduo inxqualci numcriad cundcm minorcm comparcntar iraaio-^ 
ris ad ipfum maior eft proportio,8c minoris minor- ' *; U 

1 — 9 n 8 ^ 7 » 



C vr jirtt 4 1 duo in^qudfs numai quorum a fit mMr, c t miiioriquicomfd^ 
rcntuT Ad iundcm mnortm <Mco a^dc mdortm effe frofortioncm qukm b dd c. 
fiAtrt €um fo- oddudm ^itionm,€XtTtmorum addc frofortmcomfofimfit ex 
ffcportiomifus aadbcT^^ic ntfuU purtiUirojo frofcrth bdd c patseflffcY 
itomidndc,cr pfopoiw^i AJcrorwm.crc'^wipcTMi^imMm (Smuneftoloquwrn, 
|r cmnc totum maius fitfua 'fdrte:i^itur proporno j ad c mMorifl pro^ortionc b ad 



AScfquiicxcadccima proportio;intcgrum toni dimidium fupcrac. 



r| t| gl 



u 



I I ml n I Q I p I q I r I ^ M L 



CQumra b«i»^i monjhAuit medio cxtremorum tmn fj^adobtduo ^qua diutfo: fo. 
num mtmmtinduoaqua ^cometnai r4fio»r p4rtm, cr/^^tw'iifmbwc cr<«« 
^itrinrtcuififqutfeftimumdccimam yi::/f€fqmfcxmmdciimumirctincr€ frofowi^ 
nem.hjec uao ojhndit fcfquifextadicimam frof^ortioncm qu^ illtc tx prfrtc wfrn* 
/?anu dcummifqye reiinquiturihenUtcnio.intepoquetoni dirmdiocjlemaioftm. 
a i^V^^ monjhMtjePiuififiimamdciimam ex gramorifastcfum^mmieodM 
torn dumdioefjcrmncrm.Sitcrio ut'mquintn buiusffaciumab in decern a-otlo 
rfcjttji p4mi vera c d efa-rdiquoi nom diuifumiun ut a b earum fartiucon* 
tineat decern v ^c.Cr ^ bfedccm.tr Ibdccem CT pfttm.m^mfefivm eftabcr 
c b{ut pius wfum efl) e[[etonu:V lbi'jadcbi6 ej]t[efqw[exude€mdyquam 
UfcirbdiJJciw ixeo tlfe intirropmitonie maiore. i^uomam enim fcrdecmam buiusj 



II 



^5 



I IP 1 1 i5> I M '8 I 17 1 Aadb fcfq uidccima re. proporr. 
1 d tf I c I ^7 I a I b I cotinctiqg integro co*dL'c:on.cit . 

Csiwf 4 decern O" ^^^ intaualli fdrtcs,Z7 ^ decern tj:;-f€ftem, c decern tT wowc 
£ir tt»4J<rpriwadedm4:d ueto decern ^nouem^ una odofia^perdecimam huius 
dadbejl fefquifefrimadecima.CT ^ ^d afmiiitcrffjuifeftimadecimd, funt ergo 
c 4idb dmejefquifeftimadedma adinuicem conmnd^.]cdc7 fereandim d ^d 
h (cfquiodaiia eft atque tonus. d enim continet b CT eius fartcm Q^tauAm, add 
dd b proporria maior eftfrofcrtione cad b^nam unius fan c^laua maior eft una 
(eftimadecirnaiergofefquifiptimadtcmai frofortio ferdecimnm froloquiwm mi^ 
nor eft integrofcmitonio.quod enim dufLtwm non imflet integ^rumi neque id 
quoque continet dimidiuntyifl ergo notmrifefqmfeftimamdccimamfYoj^ortionem 
i ntegro torn dxmidio effe minorcm . 

|A Cintcgram toni mcdictatcm.intcr fcfciuifcxtamdcciraam &Tcfquifc- 
ptimamdecimim proportioncm caderc ncccflc eft. 

CN4m fer4!f(*decimdm,ferqmfextadecimdmaioY eft toni dimidhtf^ fer iech 

mdmtertiat7i]^qu\ftftimddtcmd minor eft eodem toni diinidio^tfer c^mmunem 

fcientiam inter m^A atQue minus dmidioiiffum dimidxwm confiftert nccefje eft 

erzp inte^Htn toni amdirnnAnter fepiuifextiLnidecmam ^fej'qujf^ft ini0n.dtci 

D Jtj. mam 



mUtds iirJextaiedmdUmluueft^ummie decern Cr/€ptem,fdrspxtadedmd. q; \ 

ft eidctH fumm^ diUcUtturftent decern cr o^o^ (^fextd decima §4nius. dt oSio- I 

item cfo^aiecima unius di decern cr feftem:fer diffhtitionem eft fefquifextd \ 

decimd.sunt ergo decent cr 06I0 ^ fextadecimd unius ^d decern cr feftem, CT -; 

decern CT feftem dd fexdecimiduM coniunCi^e fepitdp^xtaiecima.fed dect w <^ 5 ': 

Ho cr fextadecimd umus: per fntcedentem ntdicr eft fepjuioAodd dd fededm, 
Ham decern cr ^0 ddfexdecm utfrtus wfum eftxfefqulo&oMd eft.igiturfcfquifex j 

tddedmd bu dMCla:tan$mi,tontque interuaUum tranfcendit^quare fer nonum fro \ 

loqmmfejquifextddecmd frofortio mtegrum toni dimidmm fitferdt. quidquid 
emm bis dn&wn trdnftendit tdiquid:id uUrd eius ^midium effe neceffe eft.ix quo '-^ 

ndenticre iurecognofcitur fefquiquintatn decimant cr omnem frofortionem fef* 
quiquintamdecimam cr omnem^rofortiomm ftfquifextddecima maiortm: iwrc* 
grum toni dimidiumfuferarc. 

13 CScfquifcptimadccimaiminor eft integro toni dimidio* 



V 



M 

mdm uim nuiffi efl^ui U «^ n0tpjhftip9qae Humfr^nuUof^^^OiiJtfmd 
tnonfhauitlfim contingct:ytnti]uein g€4mtetrkis iutrnHnquditdtt^dmc^cef 

w,corp\tutoqutnutm0tmm(^fur4s\ fcteft^edhoc ultimmiexidtm locor^- 
quircnd^m efl^ 

C Semitonium minu$;duobiis tcrnis in chorda fubiungcrc. 



!A cl d\ c\ "I I ^^ 



I I 



CintcUi^tuT pmiwmiifH fibmytp:qu4ndo ex dcutiori fdrtc duAu^ijuiicm to- 
nis di pauiorcm fortcm relUiis ji^frm pmitonium coUocdtur.fr ^dungi utro qua 
do ip(um ex fdrteraniffuni duobus tonis aS)ibctur.Centinuocrgofcr ficunidm 
hiiius duos wnos in chorda a b Pcrnotds acd bjitquea be bifrimus: ^fectm* 
dusifitcbd t. qui quid per diffimtianem in jejquio^laud frop<mioMe conft^nt 
^' prr itcimsm /ipdmam frimi huiuSf du^ pfqui^dk^e minor esfunt fifq^- 
tertid frofortione: erit ergo d bCT d b minus pfquitertio. Diuido ergo a b 
^ , in qudtuor f dries ^qudsicr initium tertue fi^Honis ^ fdcio nctdm f, ltd ut e b 

tres tdrum qudrtdrum cantincdt per difftrntionem ipturdb ad eb fffquitertid 
fropcTtio eft . fed /efquitertid proporth db Cre brfitperdt duos tonos d b &d b 
in propmticmd b.CT « t.efl tgitur dbcT f bperdefcriptionem femitoniumtm^ 
wusiduobus tonis (ut prop<fituyHeratJfubiun(}tim. 

C Semitonium minu$:duobu$ tonis prxponcrc. i^ 

Ta c| d\ ,c| I I I T f 

i' ._b ^ I I 1 n I I n 

CFrffW4 h dd c b ftfqmtcrtium vtterudLum CT diuido c b in odo^tquds 
p^cs. cr /wpr4 c uerfus d, fddc d c uni esrum odo pdrtium ^equdm: itd 
ut d b cdTum pdrtium muem continedt, CT per diffinitionem db CT c bu^ 
num c9nffituunt tonum , similiter diuido ,d b in odo ^quds partesio^uni 
tdrum doiungo fuptd dper notdm e. qudreiterum e b CT d b erit tonus. fimt- 
qucdu^t^mcb cr^^*pi^l^ &^^ fifquitertid proporth: maior eft duobus 

iUis 



M 16 

VlisHtds In pwpdrHow d h^ th.t^ :gitUY per difftnithnan ah & e h 
ptmt^nium tmnuf. quod cum fit diuAus touts pafofitum^t{uenUmex ^arUu* 
miffitiUis ddfun&im:fd£linn cflpoj^fiturn. 

^^ CDuobus tonis:dicfim,fcmitomumquc minujintcrpo acre. 

]A -^ dec b} 

Cidcm dUfitn 4tquc/€mitomummmus,hk($(tutmquoquediSum ejl) iwtrfff- 
gimus. Sit ergo db cr c but in fr^cedentififquitertidfrofortiointotdchor 
id db. dbd,c unfits: intendo tonum per notdm d, o*dbc uerfits d remit to 
tonim ut in pr^cedcnti fdSum eft per notdm r.itd utdbcrd bfit tonus, ^fi^ 
militer eb&cb tonus. mdnifeftum dbCT^ bjifquitertidm propcrtiomemijupe' 
rare duos iUos tmos in medid proportione dbo'^t^fft igitur per diffinitionetn 
d b o* ^ ff^fitnitonium minus duobus tonis interceptum,cr pTopofitum* 

i8 CSemitoni; mfnoris, minimos numcros rcpcrirc, & quomodo libcc 
fcniicx)nium minus in chorda tna aur pluribuj collocarc 



1 ?»4 1 


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1 


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P 


1 


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1 8' 1 


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1 c 1 


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€£ Sit d nouenanus ^ b odondnus t minimi nnmert tofd. duco dinfiyO* 
a in b,o* ^^fiCT confurgant numeri c d e. inter quos per fixtdfn quar* 
ti arithtnetkes: erunt duo pfquioHdui, duoque toni in minimis continudti. 

Vvij. ^ sit 



M 

Si; VTJt:cr€J Iqti^anxrluSy CT in tcrndvius^i^ttP I In c4eCr fr^ictottf^ h: fn 

Cr proindc duo toniJuco ffjtterea mdcm minco' frmcnuu k.pero^^t^umf^ 
cunix dfithmfticcs fdi k:c/? proportw fcfquitertU quxpa 17 ^imihuxus W4wr 
ej} j^b,^ii^<TffMf/lb k:(J;c(?a|ofrti mimmos numcros f^m\Som\ minoris.feni 
f\n:numcrifamt9nij miffmisfer difftnitioncmn^tHmcji^lid quodmbMiddidm 
dccUrdndum ejl^Hdm quU c ejuntinfij proportionc minimi: per decunAmo^d^ 
uxm tcrtij drici}tn:ticcsJuntco}icrdff primucT quUetum i CT d qiM^Hdrius tT ., 
nouoLmus/unt conttd fe primi.cr^o per undecwtdm trrtij mthmetic^ ejl, pri- 
tfiui ddc.o* pcrdccimdm ciujdem b qui ndj'citur ex I etc primis cidem humcro c: 
nu fTimusddc.Kurfitsmctb tcrndrius dtque oClonsriui: frnt dUnukcm primi 
erp per undcdmdm xcrti\iO' »« primus ejl dd cfcd CT c^*^ U^ qudtcrndrius cr 
tcnunus ftnt ctidfH priini:cr^^ per dcdmdtn dufdcmtm etidmprimtis eftddh.cu 
njo c ^m moMflrdUfrntprtmiirrioper cdndcm dccimdm tertijynummss k ex 
e cr^rrodu^s.primHi cjidd kfint iuqueh CT kfimitortij mmoris numeridd 
inuUem prim:qudr€ per uiccjimdm tertij dritbmct'ices in |w4 proportionc minimi 

Juod c/l propofttum quo dd hoc.u ddtd qndcunque chordd^fiedtn dimdesficun* 
wm rtumerum b.CT cdrumpxrtimn dccipiAs ficuniam numerum knumeros 
Jr fcilictt (cmitvnij mmoris interuMlmn^in dtuerfu dutem fddHim^ idem feceris: ft 

' e^s chordds ^tquMes aqudliter tenps unifindfque po/ueris, CT pdrtidris tdrwni 
qudmcunquc uoles fecunium \hqudTwm pxrmm dlterius dcceperis ftcwHiwm k: 
rrtf twm in pluribus chordis fitnUomj minoris couJl'Uutum intcrudllum, quod eft 
totum propofitum, 

CSertiitonium minusiinmiaorc quam fcfquifcptinudcciaia fit, pro* j^ 
porrioac confifticQuc fit vt rcguU fcmitonij fuoicndi.noa fit diffcrc* 
rutnextrcmorumtoai in duo xquaparti^ndo. 






Csint utinprxcedentiyhk minimi numnijemitomj minoris lutducentid quinqud 
mtUd fi^,Cr ducentd quinqudgintdfrx,^' ducentd quadrdgintd trid. cdpio per 
decim^Lm hm^^sifiptimdmdecimdm £4rtcm numeri k ducentorum fcilicet quddrd- 
gj^ffffriumjeritque qiutuordedm CT quiMquefipam^edecvTue.dddo'itaquequdtis 
ordcdm CT S fiptimdfdecimds dd k ^fidt I numerus tsj CT S (eptimdedccimje^ 
eriter^Tptmerusldikftfqmftptimus dcdmus.dtl ducentd quinqud^ntdlepti 

cr 



td fix.confijict itaquefimltcnium minus f(T wtdedmamhuiusiin mincrcfrofoYt 
ti^nc quawfitfrof<ntiof€fquipftitMi€chna,qudrecT^fonl<^i in minorccon^ 
ftftit ftofomotu Quamfit pfquifixtadcdma. ccrreUr'wm hinc notmn cfl. mm 
hec fado farticndo extrmorum^ani ffaciumitx quinta huius cognofcu/ntur pf* 
quifixtndedmadtquefefquiffftimadeciMa altrinfecus conflitul quarum utr^que 
fraJcHS Monjlrauit fomtonij minoris habitudinetn effe minorm, 

CSefquio^auadcc. proportio:femitonio minorc rnrfus maior euadir. 



c X } a I b 



Csiwt J h minmx numm jtmitom] minoris frdedwdtnoBiUidmhuius referti:fdt 
licet ducenm quinquapntafex^^ ducenm quddraginm trid.cdfh ferdecimam 
huius, odiUiamdcdmam partem Humeri b:qudmreferio effe tridedmO'/emis.qua 
quidem addo numero b^fCaque d^egatus c. tunc cadb fefquxo6kMiddecima pro- 
fOYtio eft.dt c maior Humerus efld.nam a duntdxat continet ducenm quinquagin 
tnftxif uero^qualiumfdrtiumtotidemxT infuper fimijfem unius. ejl ipturfef^ 
quiodaaddedfHd (roi>ortio:j€mitomo minore maior* 

^j CScfquiponadccima:cft fcmitonio minore minor .Quo fit vt femitoni 
um minus inter (efc]uio£tauadecima,5c fcfquinonamdecima confiftac 

collocatum. 

1 ^^^ I ^i^ ^^ f ^43 _ i 

I A I c 19 \ ^b I 

iLsint ut frius minimi termini pmitonij minoris d^ b: dico fefqulnondmded* 
mam ^o^ortiontm minorem effe froportioned dd b.capoenim ^^dedmamhu * 
iusinoHdmdedmdm fartem numeri b^quam inuemo effe duodedm cr quindedm 
nondfdecimds.qudm fdrtem adiido numero bicrfi^^-f^op^o cadb ejl fejqui* 
nonadedmd . Jed d maius eft cA^tuf fer wKdedmam buius, nudusefl fimitonium 
minus [ifquinonddecimd froportioneydtquefejquinonddedma profortio pmitonio 
minore minor. X^miarium ucro ut decimaquartd huius inotmi effe pofe/f. 

J 
^^ CScmitonium maius:in data chorda conitituerc. 

E sin , 



J^ 



M 

r:^ . I ci II \ I I 1 I ^^jal 

€sit dbcr^*^ [etmtvnij minoris interuaUum, diuido chin o^o f^artcs aquasi 
tjVibus ^qudm fjcio fOTttm c d.itn utdb fioucm continent, rrgodb^ ch crit 
tofiusJemo i^itur dtonodbv^: bftmxtcmum minus fdlicct abcTcby rflin- 
qutturque dbcr^dbtom uli^udfdrsa-q^^iff^pfniton'mm minus fupcrat. 
igitur ^n diffimtioncm d fc cr^ fc/fwifo«i«mfn4iMi (j?in ddtdchordd(utfY0fO' 
ftcum adt)couj{itutum,z7 quern udmodu^n pmitomum maius coUocajli ad fdr- 
tcmgr^uiorcm ,itd qu0quc dd['drt<m dcumims coUocare fdciUimunjffftt. v 

CScmitoni; maiori$:minimos numcrosrcpcrirc. 2.] 





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^Sintdb minimi numcri pmitonij mincris €x dccimdo^dUdhutus rcftrti CT 
foCiomnus. quonidm ix dnnon^rdUom duim^toCkaa^xhuxus f otlofuyiu: pi 
mus efl ddb.ergo fnon numerdt b. non hdbet iptur b cddi^dm fdrtan , Vuco 
igituffinbo'<iV<^o^f*^S^^ ^ d. fer pftim^M licundicUmcntorumayithmei 
nets dddc\ut dddb, {jligiturinterdcfcmitomum mt7ius,fetmtoniiqucminO' 
rti intcrualli^nt , c;- quid chdbet f^rtcm c{ld4^dm ut b: dcdo adftn r.umcroc 
fdUcm o^lAUdm ciui b: CT codccrudtus /jJt <. continct igitur enumaunic^ 
^ p4rffw cius oddndrr.quare ec tonus . CT 5«w ^^^i*^ f ^ Juprrdttfimitonium 
minus d cinteruallo e durgo t d furX numcri pmitonij mdiojis.QuU dutem 
iidcm fmt minimi: fddlccx dcciindquintdtntij drithmcticcs coptofcdJydiJltd* 
Ixndo ddb c yCJ' q^of^ rdiquum fucntyitcrum quoties fotcs dtflrahcTido dbd, 
Cr hoc VdClo dcmccffs . CT uidcbis dd ultimum rdUldm umtdtcm: eritque modo 
qui dd Idtus dfpdret dijhddicqudrc fcr edndnn dcdmdmquintdm: d CT ffunt 
ddinuicem frimi. funt ipturj^cruiccfimdmciufdemtin fid fro^ortionc minimi 
quod^cfl' frofofiturn, 

CScmitfoDiimaiorfs habirudo: fcfijuiquintjmdcdmam fuperat proi *4 

for* 



portionem. 




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€sint a h mlmmt termini hdbitudims fimitonij minoris per frctccdenUm repcrti: 
Jiicohabitudinem aadb maiorem ejfcfifquiquintadecima.lumo enim (ut in f race- 
dentibus iayfdefcfdSlum eft) per decimdm huius:quintam decimam partem nu 
merib, Cr hoc Inueniturxentum tri^intd fix yCr o{loquint<ed€cim^ wniusx que 
fit d. Adiicio ergo d adbcr fi^^ cicertum ejlcadb ejje fifquiquintamdecimam. 
ate minornumerus eft quam a. eft enim a Humerus duummillium centum o6\u 
dgintafiptemi c dutem folum duummiUum centum o£iuaffntdquatuor CT f^^ 
fimis pMiloamplius.iftiptur per undecimam huius: fimiton^maioris habitudo 
fifquiquintadecimaproportionemaior^eamquefuperans^quodeflpropofttum^ 

jy C Aporomes interuallum: minus eft fcfquiquartodccimo interuallo* 
Vndemanifeltum efl fcmironij maioris proportioncm; inter fcfqui* 
quinramdeciniam & fcfquiquarcamdccimamrcpcriricollocatam. 

I 2194 4 I 2187 ) XO48 I 1465 4 I 

I c 14 I A I b I d 14 I ■ 



C Apotomcn ^r fimitvmum mdiusiidem effe lam diximusSint ergo a b minimi o 
numerifcmitonij ma\ori:> ut duomilia centum CT oduagintafeptem, ^ duo wtlid \ 
quadrapntdodo, fumo per decimam huius quartamdecimam partem bfcilicet^^^ 
duum^ milium e;x^y^dragintao6lo:quaminuenio ejfe centum quadragintafcx^^ ^^^ 
quatuor quart ^decimayqu if fit d. addo earn quartamdecimam ad b: cr fiatnu ^^^ 
merus aggregatus c.tunc numeri ad bfefquiquartadecima proportio eft, atcma^ ^I 
lor numerus efta-fiquidem duomilia quadrapnta o6iOy^ centum quadrdffnta 
fex^O- quatuor quart^decima: fummam ftmul attoUunt c^duum milium centum ^ 
nonagintaquatuoTyO' ferequartam unius .cr dfilumfiimmu eftduum milium ^ 
centum oduaginta fcptem.conftat ergo per undecimam huius fefquiquartamdeci^ 
mam proportione maiorem e[Jefemitonio maiore atque propofttum. CorreUrium 
ut pntcedentium correUria notum eftt 

N- Scraito^ 



M 

CScmitonium minus atcjac fcraironium maius, ia fupcr particulari zd 
proporioiie non cadunt.lcd ca m fupcrparticncc rauoae confillcrc nc 
ccilc eft. 

CiSdm ftmiioniummlnus pn corrcUrium ukefimi€prim<ehuius: cddit inter ftf- 
quioajujimdccvndm cr M«««<'»^"^*if^'««<w- Atquiimr fefquioaMidrndea- 
mxm a- ft^quinonaMuimjiminuiU caJrrf uAct macct^U, medUque /kpfrp4r 
tuuUrii h^itudo.fuHt cnim ilUftferpdrticuUres proximo maiotdtqi^e minor, |- 
giturfimironiumminus:infrpap^rtUul^rdtion€COhffltt.nequc fcri^m pmi- ^ 
tomum mdus.^^m per carrcUrium pr^cc3entis :<jdit in Aiqua fropoftione me* 
du inter lefquiquintatTtdtcimdmO' fefquiquarumdccwumduaifroximdsjufer 
fartkukr€s,qi^ fupcrpsrtkuUrc mcdum nwflum admittunt.noniytur fctnito^ 
nium tHMUs : CAditinfupcTfarticuUri rutionc.fed V ^'<^« rdnomi{mxton\j mh 
noris atquc mMons . mulris (u per particaUr thus (ut iam uifum efi ) funt minorei 
cruntftum minorei redone dupUri, qu^multipHcium mmirnd eft, utqucdmini" 
mo numcro denomincturnuUus cnim numerus hinAfio minor, non tft er^ofcmito* 
»ij minoris hMtudo multipUx:fimiiiter mque jemitonij mMoris, re/inq«imr rgi- 
ttffdfujficienti dim/ione^ cum tile ftnt inter mAiorem terminum,^ mtnorem: eds 
jr cjje injupcrparhenti^encre, quod ell propofjtum Ad umen propter jdphiftds ad* 

'itertereiicctiquod tonus c^fujepjrtesyConloKjntl£:a- iO'tjondnturum paxtesin* 
ten[^,(emper injuperpjrticuLxri^i4perpjirhenti,^int multipiici hMtudtne cadunt* 
rcmilfauerounikperp^rhculjriJuperpjirtienti,aut[ub^^^ 
fjl: folemus tMncn eds omnes diccre ejfe in fiperpdrticuliri habitudine/uperpdrti* 
mti,M4t multipiici, idem fr per pdrticuLre CT fuperpdrticuUre rcputa^tcs: er 
fro unocomputdntes.fmdttcr jupcrpdrtieas ^ fuperpArnem ,V multiphx ^ . 
fuhmultxpXiX. 

CMuficum comma in chorda rcpcnrc. 17 

: — \ — T^\ ? i c ■ bT 



1 A I t I — 



CTsi^ d b chorddfuprd qudm fit propofitum reperire muftcum cemmd: m qud db 
Cr c bfttfifquio^ldiM propcrtio atque tBnui.fmt pr^tterca de minimi termini je-- 
Tvitonij minom.diuido fpdcium d b inpdrta ^qudles fecundum d: ex qudrum 



numcro 



numiYo 0.1 if Jo b ucrfus dyCHfio fhimium mmtrmn e, O" intcrmino edmm f o< 
no notnmf.tmc qu^ pofoitio d ad ticatrit a badfb.quareab adf b firmw- 
mum tnims.Kurfwm c bhaciwm fuo \n ^sequas partes ficmdntn rmnaum e: cr 
eamm fartium ab iffi b utrjus a metiendo^ funmo fumidumitumtrum d^er i« 
earutn tatmno fono notum g.mamfeftumetiam ejl gbcTcb tffc feymtomum mh 
Tius.fed cir cum a b erf b fnrobatumftctiam pmttsmium tmnus. ergo mttrud* 
lumfbcrgbefl quo fefquioikiua frcportio d t CT c bimaiortfl duobus pnut^ 
mis mntoribus. eftipturfcr diffinitiomm fb cr $bmutntum ntuficmt comma: 
quod a At mmjlrandutn, 

1% CComma:in minimis numcris conftituerc. 



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Csi»t 4 b mittymi tumeripmianij muorisMco 4 m/? ©■ in t,er binficr tx* 
ursantcdr.truntqufcd crdeduo femimia tmnora coniwnihi. fit vr^terea f 5-31441 
oaon<mus:qui ex decimacOaua hum </f pimus ad b.quare fautidecimam ttu 5 i4i 8S 
*H ortthmetices, ©r prmus ad e. von htbtt iptur e oOauam.duco ergo foilona- 7 if? 
mm m cde-.cr froueniaKt shi,ut ifdUcet oriatur ex f in e.certum eft etiam in- i 1 1 9 
trr g her h 1 efje duo ftmtonia minora, crquiae eft oOaua fars i ; addo errilob 
fwiul,fttqueeorum aggregatus k.dubium rti'iUum eft kadi cjfe fefquio{htuam-& S^7 
k gejle frofornonem,qua fefquicihtua maior eftgh a-h i ducUs fenUtomis mi ^69 
noribus.^u^tigiturk gmmeri commatis.fid quodfintmimmiex decmacuina zfS 
fm<i ^'fl'waif « mtum eft. nam fulftraao ut eaiffa docet gabk,^ eoquod 1 1 
retiaum eft fuhjirado quoad foteft .,b g, crftc deinaps: andem(ut ad lotus ad- S 
tci{umapparet)rrlinquiturumas. it quemadmodum comma duabusdiefibui ell i 
\r^\ytum:itaquoqutqukmfacillimum eft comma duabus duftbus rubiwteerc 
tutduabuiinteijtrert. ' * ' 

"" ^ 'H Scfquifc- 



M 

C Scfquircpruagcfimaqaarta: commacis prcport/onc trranfccnditur. *^ 

A 1 d 74 I b" I Z 74 r 



€sint enima b rfunimt commat^s, f^rr fr^adentetn ref^crti. tliciofcr dccimam 
huiuiifi^tua^cjltrtdmqudrmm b,qui€ (it c.ddtiiio Itaquc hcTcfimuh ciT cojUfatt 
a^re^eturquc immcrus d.quifi ad b confcrdtur:compaitur ^fquifgftuagcfmiuf'^ 
quarius.atqui idem d minora cj]cdcf>rehendJtur.fuftrat cr^o ptr un^imamhu^^ 
ii4s:rMio commatis fe^tuagefiwamquannm frcfortioncfTi. 

CCommatis ratiorfcfquifeptuagcfimatcrtia proportionc minor eft.vn^ 
dc fit vr commatis ratio : intcrfcptuagcfimamquarcam S^fcptuagefi- ^^ 
numtertiam conftitura rcperiaturhabitudinem. 

I 5^'47o i I 53'44' \ 5^4^^^ 1 7iSz i f 

I d 73 I A I b ( c 73 I 

Cswf d t,Mt frius,m4nim4 mmericommatis . capo feftudgspffiawtrrtum fdr- 
f^m t:^«rf /it c.iwff^ t CT cf\md:V ^SS^^^^^ dcritque d dd ^ fefquiffftud- 
gtftnMjiertiuj, CT d maiotd eff^onff kit ur. ergo comtftdtis rdtio ferundecimam 
huiitsyfifqmfeftudpfimdtrrtid frofortionc trnnor tfiiquod erdt mortjhandumXou 
rAarium ut did notum tjt, 

CCommaris ratioiin fupcrparticntc rationc confiftir. 



CNa?t cnim in fufrrfdtticuUri confUet : qudnioquidem diut fuferfort^cuhcres 
froxiwif fcfqupftua^ftmdqudTta^ftfqmfeftuageftmatertia, omnan frorfus 
medum excluddnt/upcrpdrticuldrem.CT ^^^^ mnus ingcnne trndtiflici conffle* 
re udUbit: ut qu^ fe^uagintn duds fiprrpdrticuldTts hdbeatfc mdiores.relmqui0 
tut ergo ut cd injupcrpdrtiatte^cncrc conftjlcrc fojfit,eft enim ca mdorii CT ^- 
norisHdbitudo- 

CRationcs fchifmatis atquc diafchifmacis: funt ignotc,atquc irratio- 
nalcs. Quo fit vt quarum minimi numcri tctragonicum latus non ha» 
bcanrracdictarum rarioncs ignotc irrationalefquc fincomncs. 

XS6 



I r 

I'' ^5 ^ I -^43 I I I T ~ 

I A M b I c I d I c I - 

€L^ds ut'ioncs i^noms atque irr^ondles dicmus: qaa nuUo certo^coftjlitutoqtdc 
mmero difignnu ualent U7iquam.ut ncque in giomctricis diametri cr cojle qua- 
tlrafi poporno; quaUsfchijntutunty diafchifmatumque ftofcrtioneseffe dicimus, 
swt trgo fftmoab minim ^mtowiTmnorisifcr dccirmmo6hu<tm huminutn- 
tiMemm^ut \Uic quoque^monfiratum eji) nafcitur ex nouenario in fc^^ temario 
inprodudum noutnar^ m /c,at Humerus qui ex duSiu nouenarij in/e cxurgebat, 
erut quadmusizr temarius nonefi quadrat us. ergo per decmamquarUmfixti 
arnhimtices b mnus femtonlj mnons extremum^ex duSiuquidem non quadra* 
ti in quudfatum poueniens : non ejl mmerus quadratus . ft foffibile eji ergo ut 
diajchijmatos j roporiio in numens fit nom ifimergo fcrfexmm quarti ariibme-^ 
ricfi dwo diajihijmatn in mnimscdt coniumbi. m^ifcjlumenmefiawidia- 
fchifmajit femitvntj mnoris dwiidium icd^-d efinuil effe femtonium punus, 
a- cade cjftfermconij mtnons interualhm. fed cr cun frofortiones cd O'de 
jm comihUAie in mwmis : it go fer quintnm quarti arithmetices^c e fumt in fi<4, 
froportione mtnim.funt ergo mnirm in froportione femitonij minorisfedcr^^ 
in I'cjiti Junta C7" b.igitur c cr e iidem awnt numcri cum a cr bifciiuet c idem 
rf,C7 « idanb,Vt<£terea quiaqua i^roportio c added cjl dad e: ergo fer f>r imam 
pxti aritbmaiccs c mmous ij\ quadratus ^^ e mimerus quadratus. quare gr b 
idivt nurntro i etiam quadratus, at b demojlratus eft non quadratus.erit itnqi idem 
momrus quadratui cr non quadratus:qhod eft im foffibile. non igitur diafchifnta 
note frofortionu haUbicur.tt dimdefrofortiontfcklmatosdtmoyt^rabttur^fint 
cnim a b fidbter dejipiati rrunim numeri commatis. quia fa uicefimamc^htuam 
huiusb ?mnor fro fortionis commatis tLrnUmis:fit exdu£tu o^onarij in quadra* 
turn iateris ducenwrmn quinquagintnpx^XT o^lonarius non eft quadratus : ergo 
fcr dcctntdmquartnmfexti aridmetices b ?mnor termims commatisy non eftqua* 
dratus.non igitur yfimiii utfrioris fartU demonftratione : a b certis defgnatifque 
numeris foteft aquis frofortionibus diduci. (ft igitur fchifmatos ei us jalicet mea 
ditmtiiratio ignotayatqueirrationalis.QorreUrihm ex wododemonftratiotds no- 
turn eft, 

53 CTonuj.duobusfcmitoniijminoribusrSccommatc conftat. 

CN4m rdtio fifqui^huaiduobus (etmtoniis minoribus y atque uno cemfftd* 
te conftat.jufaat mm duos dicfesduoquc firmtonin minora uno cotntnatc. at to- 

E liij nus. 



1 



M 

bus ^commatc. 

CTonuj a duobus fcmitoniij minoribus:vno commate diftat. ^^ 

€:Sdm fubjhd^lo d frfquioifhudfrof^ortione unocommdterelinqmmtHr duo fc* 
mitonU minerd: igitur cr ^oicm commate d tono didu^o dua Jwfb C7' duo fcmi- 
tomd rmnoTdrclirtqurmtur. dtjldt cr^o tonus d duobus femitoniis tninonbusiuno 
commate, . ^ 

C Scmitonium minus rribus commatibus maius eft : vcro quattuor. 5J 
yndc manifcftum eft apocomeniplura quattuor &c pauciora qumcp c6# 
tincrc commata. 



q 5 lb 1x63^9443^5^35^^^57^05 7/^^^ ^76 I 

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n ' "- —T" " xin<gP35><^3Q7 555776^3^o747 

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CNo« efl er^^ curiofitns calculi Ubore dctcrrit:i : quo minus quotcommdtn in 
dufi quotwdPotomcyQuot dcmqutin tonofmt^ fcruejhgaret. quod n\ftd friom 
bus trnamm copomlfofhcum id quoquc flus Uboris quM^ut mjhi uifm c[l) in 
muficis moduldtwrnbus ufuSyUCilmttfqut dfferatiMijJum feajfem.qui mmtn id co* 
gnofccrc dtfidadueriffihoc {d^o dej^rchcndcnt.Sint d b mnimi numeri fcrmtonij 

Tfunoris, 



1 1 *1 

mifmlSf tr^i ffunW ccmmMs : jvr icci/nidme(hudni & u\ccfm<mc(huAm 
hums reftrtx. duco t m c CT" d CT frouenutnt ef,CT dinc^- ueniatg: fcrfiftU 
mam (ecu/ndi arithmeticfs fade eft commatis habitude. ^ fer c£ktuam ciufdcm: 
gade habitude diefcosyfif?Utcnii<u4€ tnhtons.T>tindc duco t \n f, Cr f \nf^ o* t \n 
g:Cr ndfmntur h k l.pcr fextanf^uartiafithmetices firqudm facilh cogncfci fe^ 
teft k hicotttinnc duo commataicr fnpptimatn ftcu/ndt eiufdtm I ad h ejpfimi- 
tDnium fTunusJdnde duco d in b,tr c in k,cr d in I: cr « ordinc ucnidnt mno. 
fer eandem Rxmm quarti cognitu fdcillimuw eft n m continere trld comntdta^O' 
ferjeptintdm/ecu/ndi: o m continere jenutonium vUnus. at n numerus cogncfcitur 
effc minor o.ergoo dd m /e/futonium m?ius:triduindty exuferdtque commdtn. 
Deinde duco hinha-^in k. crhinl. crjuo ordine exurganty orianturque f q f. 
mdnifefium eft fer idem quod friusiq f continere quatuor commatTiy^ r p contis 
neYepmitonium mituis.dt numerus r minor eft namero q.igiturqudtuor commd* 
t;t:dmfUus funtfim^tonio minorccorreldrium dutem hinc notu/m eft:quodftmi* 
tonium mdius Jolocommate Juperat ftmitoniummdnus^dtqui/emitonium minus: 
flurd tribus O' fdudordquatuor ut modo uifum eft,continet commdtnjgitur w«i- 
co fitperddie£lo commate : ftmitoniwm maius quod uoaxnt dfotomen ^ flur4 qudt 
tuorcr fdudord quinque continere eft ncceffe. 

}6 CTomimrplura feptcm continere commataneceflc eft. 

€f9dm tonus ex ftmitonio minorc ^ dfotomccodle^cit dtqucconftitmtur.dt ftmi 
tonium minus fer fenultimdm tria continet commatn cr dmflius: ZT f^ fr<fece^ 
dentcm dfotome qudtuor ZTdmpliusAriddutem cr qudtuor CT dmflius zfiftem 
funt cr dmflius Agitur tonus flurd qukm/eftem continet commata, 

€,secu^idi elementortwt Muftaditwtfinis^ 

Efquitonus : eft qui tonum ac fcmitoniom minus con# 
rinct, quern & trihemitonium , trifcmitoniiimquc : infe* 
rius dicemus. 




Confonantix fimplicej fimBiate(raron5diapente,& diapafon. 

Compofitaf^erordiapafonjiiapentejbisdiapafon. 

Diarcffaron : cftjConfonantia,qux ex intcrualli fefquitcrtia ration e 

nafcitur. 

F Diapcntc* 



Ditonus.-eft qui duos complcftitur tones. j 

JTritonusvcro^itres. 1 



i 



y 



M 
Diapenrctqusnafcirur cxferqualtcra- 

Diapafonvcro-.qiMCCxdupli. .- r 

Dupafon diapcntc: cftquam adinuiccmiurtaaf conftituunt diapaion 
ac diapcntc confonantix. 
Bis diapafon:cft quam coniungunt duaf diapafon conlonannx- 

Cnvc d\uus dfcendu^t Vytlyagomv.qucd altius afccndcntihus uoccs quoquofd^ 
ao iIlisJlTiduU uip€jtnt. C7- quodumcuiquc fame fu^ uocis moduni, iw^itefque 
ddcor/fonamUm bn du^afonnaturdfcccrit.quodquc habitn contcmfjgionemuft 
i^tadufqutconfcnanti^ntbis dU^jfon : reliquam uthabcatur quhm fjdi/pM^rt 
fkmu€n0it^ut qui ad tir ntquequatcr diaj^a/on niufuos modulos af>tnr€ wolttf- 
fint, cr h^c quoque de auift nuifici fcpne cmncs in dcfinicnda y Jeter minanda- 
que atqae tradcndu A/ciplmrf rmficn limites VythAgore non tranfcfndwfft:f^ut^n0 
tcs eius Ihmtibus contentiyV ^faifTiyueteremque Mithorimtem jecuti, fuffidetet 
detcrminaffe.quod CT nos in hoc opere tennihimus imitnri. 

CScfquitonus.inrcr fcfquiquincam atqucfcfquifcxcamcollocatus cO. i 
vndc fie vt ctfi fefquironus iocundx,fuauicerqucaudirum fenat; non- 
dum tamcnconfonantiaponcndus fit. 



I 191 ^ I i8!^ I x8^ 3 1 156 I 14? I 4^ ? f 40 3 



I f H g 



Csir er^o a ducenta o{lu4pnr:i o{lo:b una duccvm quinqua^nt^feXjO'C ducen- 
m quddr^pnm trid.\^erdecinuono6hiumftcumdi huwyjn^nifeJlufM tjld adbef- 
fe tvnuntj^b ndc effcprmtomufrt mnus.quarea ad c^er dtffinitioncfn cnt [(f- 
quitvmis . quern dicpin frofertiont mincre confijlerey qudmjtt fefquiqumtn: cr 
mMore pfquifexta. Hamczipc pa dccwu)n fcmr.di huim qumtam faiUm c:p* 
aen'utm4merus 4$ crtres quints qui fit fadd0i$iturfdd c cr a^e^tus fiat 
i: qui maior inuenityr qudm d. \ptur prr undecvnam (ecuftdt htiius:dad c pro- 
f orfio mdor eftqukm 4 ad c.Kt Hero d Jid c [cfquxquinui cjl^eft imque fefquitonus 
in froportione motore covpitutus:qudVi ft fefquiquihtu proporfio. Capo diniqut 
\Hr ejftdem decimam ficundi ftxtnm partem c: cr wtw^ff ^''i^' numavs 40 CT 
pmis qui fit g. a^e^tus jgitur ^ad numerum c:r(f\ituat numcrum e.certum 
tjlfumerum ctmnon'm effequkm a. quare ut friusydad cfefquitr^jus: md:0Y e}l 
qukmeadc,quiinpfquipxt:Hy7CpcrtioneconJ}ituitur. quodcfi propo/iiKw. Cor^ 
nldrium etiam copjofcitur.Q' pimum qucd fefquitvnm fuuitcf feridt duditurm 

cuiufibet 



tuluflibet nmficls nidduUtlonlbfHinuntiyfidentfacitftnfiS.quoi ucro noniti con' 
finantUftt : iccirco euenit quod fefquitD?uii in jiperparticuUrirdtioru non confv 
ftit,quandoquidcm inccr pfquiquinmm CT f^fquifexmrn froximdsfuperparticuU' 
r(s:nulla cadit interJlcs,mcdidquefuf€YparticuUm habitudo. ncquequUemeft in 
pmlttplicigeneretqyomdmfcrumecinum frimibuiusdufla frofortioquar?id'- 
tipiicium minima cji^x fefquaketdjCr f^fquittrrtid froportionc maxirms quidern 
fuperparticularibus, cxurptdtque nafcitur. confonarttiA autemomnis per diffini* 
tionem in fuperparticuUri out ntultiplUi rationeconfiflit.efi ergo tot urn propo^^ 
fitum notuvf, ^ 

X CI tidcm ditonus,intcr fefquitcrtiam atquc fefquiquartam racdiusrmi* 
nimc muficamcomplccacqucpcrficit harmoniam. 



1 8i 1 ( 


1 8i 1 


1 8o 1 


I 7^ 1 


64 1 


1 *i i 


1 16 1 


1 U 3 1 


l-A 1 


1 t 1 


1 B 1 


1 c I 


1 f 3 1 


c| 



Cuarmonuim c^conpnantUm idemdicmus: ^huius ut pr^cedentis procedit 
demonflratio.sint ergoabcduo toniin minims conftitutiiut Si,72,64.dicodi? 
tonum a ddciconfjUre in proportione mnore fefquitertidy ^ maicre fefquiquar* 
fa,cr fft^fii^f^ confindntidm hdndqudquAmperjicercCdpio tnm tertidtn pdxtem 
c per decmam [ecM^idi huius^ut ftpefd^um ejl : ut uenitMwm C uiffnti cum 
triente unius,qui numaus fitf^dddo imquefxiZT trientemdd numerum c 64: 
^dggrfftrrttif/^ 8$ cum tertidpdrte unius quiidem jit d* mdnifeftumefl dadc 
effe fefquitertiumlt dmaior eftd.continetenim d dwntaxdt unum c oCludpm 
tn:d uao o^ud^tnquinque o* dmplius . ejl ergo fefquitertid proportio: ditono 
ntdior-U rurfum mp'xo quartzttn partem c qu<e fit g: quam dddo dd c CT fiffgdt r, 
quxerit ddc fefqtsiqudrtus.at dvtdior ejl e. igitur per undecimdm fecwndi buius: 
ditonus fefquiquartum trdnfcendit, CT cm inter fefquitertium^fefquiqudrtum 
mdlus aiddt/uperpdrtkuldrismediusjncquevuiltiplexzcnt ergoc^tomis in pro- 
portione fuperpartiente coUoattuS . qudre mufaim confoudntidm (ctfiin mujicis 
rnoduUtionibusftteupbonus fuduxtirque dudituferi€ns)nondum tdmen perficit. 

[ CDitoniinterualIum:fola fcfquitonumfupcratapotome. 

€udm fefquitoms unum tonamcontinet integrum: crf^cui^ditomccntinctfe* 
viitonium riwms.jed cum tomis ex femitonio minore ^dpotome conjieteud^ 
ddtque codlitus . ergo fefquitono ad fecwndi toni compUtiomm : jbld deeji 

f if 4porowe. 



M 

dpotcftts, at ditomis (qIos duos inc^lu^mts yintcp-ofque foffiiet tonos.etp ditoyA 
inta-uJlMmj (oU dfotomtifoli^utfamtom mdiare, [(fqmtont fu^atintcruM 
lum.quod cfl fropcfuum* 

CDiarcflaronconforuntiam.'in data chord i collocarc. 



ji. 



t>i 



f 

CC«m enbn efinitaJ(fquitcTtiaqu€^o^9rtio,c0»pnannM7t duteffironcreeXik*- 
circo data quacwnque coorda «f J t eaininquxtuor iCquas fcrtioms diuidcutii 
c,c d,d f ,cr c t.cr dico abadc biconfonart duuffaron. Sam a b contmetcbticr 
ifi(ufcrac,qua tatia purri c t rf^w^fwr.r/? er^otntcruallum dbfejquucrtiumdd 
c b.crp abddc b^cr difftniticncm con/onat diaiclfaroniiij' conJonuntU dun[[^ 
ton in chorda a b duts^c^llQOitn.quQd cjl fro^fitum. 

CTriconusxoofonanriara diatcQ'aron tranfccndic. $ 

- CN4m ^ dcdinamoihudm frinuitres fcfquio^Ui£ fro^ortiontSydmflius fuvt 
fcfqtfitrrbo intnuallo. atqui tn tnbus frjquiodiMs: pir di|tintf iowcwi trcs confu 
fiJtntOHU rgffin- in tnbus fcfquiQ^Auis confijbt tritorMs.cr w c^itritOyfefquitn* 
tUqw mterualloiconfijlitcQnfinaHtia di4tefJaroH.iS}^ur con[9nantUi?i duatcjjaron 
tr^coidit^tfroponebaturitritonus^ 

CConfooantiam dmcflaroQ ; duobu* ronis atquc fcniitonio minore 4 
coafltrc ncccflc cH. Vndc facile comparatom cftfclqmtoaum cono, & 
diconum femiconio minorc cicra diatcflaron conccntura dcficcre. 
Cotnpcrtura item cftconfooantiam diatcfl'aron-.quinquc diclcs Scduo 
conimata contlncrc. * 



CDiate[firon conjonantia. 



Csit d t CT c t cemjdnjTttia dustejfaron, dico cam duobus tonis CT fc^fo^io 
TTunoTf conftarc. Sam cum abcTcbfit didtcfftron initabO'^^?^ conucrfio'- 
%cm drffinihonUfcfqultcntusM cum ^cmitwimmi:^ diffim^onefitid quo 

fefquitaM 



'' 2? 



fifjuitert'u. iJuohus tonis mdxoycfl.ccmiimtergod bcrch: pmitonum mhms ^ 
duos tonos.igituYConfinantUdUteffarontduobus tonU^pmitomoque minore con- 
Jidt.ttfrimum corrcianum:hinc facile co^ofcitur^Cum tnm fefquitonus foUm to 
mm cr /emiwnimn t^nus contweat dtefi igitur /p/? ad confinmiam diateffliro 
comfUttddmperpr^/entem muftonas.it cum ditonus folum duos comfUad- 
turtonos:dtcYitiffidd tundcm comflenddm (imuonium minus. Secundum uero 
kifichaud difficile perfficitur cognitu^.iiam cwni tonus fer trie eftmamtertiantfe 
cu/n£ duai ^efes er t0tum comma contineati duo toni quatuor diefes cr iuo co 
tinebunt commam.dt fetfr^fentem diateffdron conjortdntididuobus tonis mtam 
diefm fiiferaddit.contintt igitur confondtitid diAteffaron: quinquc dielesatque du$ 
commata.qu0d efl tot urn profojltum^ 

CQuinquc toni;dua$^diatcflaron confonanrias vno commate vincunt, 
' cuaduncquc maiorej. 

€?utauitAriftoxem4s muficus iiatejfdron cotifindntidm duobus tonis ^ integro 
fimimnio conjUrc^O'^oinde duas didtejjkronconfindntidsiquique t^nos implere 
cuius error ex tertio mufices diui Seucrini Boetij^o' ex hdc cr fr^cedentem confix 
nantid diateffaron mncx duobustomscrfetnitoniominore.quodexulcefimdpi' 
md fia4/Mdi:int€r (efquiodMiamdecimdm cr pfquinondmdecimdm froportionem 
coUocdtur,inte^wtn dutcm fimitonium per decimamquartnm eiufdemiinter fifqui 
fixtamdecimum ^ /ifquifiptimdmdecimam coUocdreturdiminutius ejl ergo fimi* 
toniwm minus tntegrofimuonio.Quid ergo^ut perpr^ecedentem diamn efl confix 
nantid didteffaronducs tonos ,^ fimitonium minus continetidu^t igitur diatef, 
firon cotifindittiacontinebunttonos qudtuor, O' duo Jemitonidminord.^qui^ 
d pertricelimamtertidmJecu7idi:tomus continet duofemitonid minor d^ mtum 
commd.ergo duit didtejjdronconjonantiacqninquetonos wto commdte mifiusyco* 
tinentquinqucigitur toniiduas didtejfaronconfiTumtius imo commate uincunt at 
que euddunt mdiores.quod intendebdtut. 

8 CConfonaatiam diapenccrinaflignato ncruo conftitucrc. 



[_3 [2 I I Confonantia diapcntc. ^^ 

U |c Id I b 

Csit dlfigndt^ Humerus dbfuprd quern iuffwm fit confindntiam didpente coUo 
can .diuido dbin tris ddinuicem aquds pdrtes per notzts dcd biitd utdh tres con 
tine^iO' f ^ ^^^ contineat iuds^rit per diffinitioncm d badcb: bemioliu^^ , 

Fit/. fcf. 



J 



/ 



M 

ffjUjltcrum<jLc wUruAUum.fd cctifonantU dU^tcpa iifftnltl^nvn txtd \w 
ur.ijl^i rjfionf wtf/cjfwr.n-^odtddctcow/Swrftitiiwpfntf.mtqttf 4fc die bin 
dji^ chord J d^i^ttxtoue neruOyCon[ofuntui dUpcntc coUocatn. 
CTrcs conixonlonancia dupentc minus funr,& quaruor:caadcm c6- ^ 
lonantum twnfccndunt. 

CLr;f ex qumrn huiusfatu cognofdtur tritonumnon fo[fc cfficere dUpcnte con- 
joTuntum: hjtc etUm ojhndit trixnum dijftnU confinantiaefft minorem, ^atn 
ycr dcctmjrtioclM4itn frimi huiusitrcs fifquio^Mii minus funt fcfqualtcro inter $ 
M ju'o.cr per duvnjmfionjm ciufdemiquaruor pfquhCldui fcfqualtc/hm frfirant \ 
vilcrujillHyn.conj3ndnU4M4tcm di^pent^.in (ifqHaiterofiUefl^ergotrWtoniintri 
hui fef<juio^.uds cvnUituti.minus junt confonantia dupcwfc-CT quatuor teni in 
qujLtuor conftjlentci fefquio6lanis:cdnd€m cvn/oTjantiam ma^nitudine tranjcen* 
dunt.quodrft Wtufft ffofofitunt, ,^ ^ 

CConronanriadiapcnrc.rribustonis/cmifonioqucminorccoftat.Quo lo 

tic vc a diapcntc fubdudo tono:diatc(faron confonanria rclinquatur, 
lubduda autcmdiatcflaronconronantia-rclmquarur&: tonus. 
^sjm pcrdecim4mquintdtn*frimifi af^fqudltcro intcrualio pfquitcrttum demp 
turn tuctit:rclirtquctuT f(fquiodjuum,/id ut in danonfltdtione fcxtx huius ui* 
fum efl:ftfquitatiumcontinctduostonos cum /cmitonio minorccrgo confondntie di 
- dpente fefquioad44um hocejl tonu,ultrd duos wnos cum femitonio minore contina 
tubus toms cr ftmitonio minor t conjidbit qucmddmodum propo/ituw (^.Correlx 
rium cognofcitur.Udm didfctcpcr fr^fcntc cSttnct) tonos cu fimitonio minore 
dtfubndC\otono:refidui/kntduo^wni,dtq;(cmitoniumiru4s.crT^r6buius x toni 
cu femitonio minmicojhxuunt cofondntidm didtejfdro.ful^d^lo igitur tono d co*^ 
fonxntid du^tetreliquitur dutejfaro.fcd cr cudidpete coflct ex tribus tvnis cu 
femitonio miner eifidfjhdad ergo didteffxrS.cofindntid q^ duobus tonis crj^^^'^^' 
nio morecompletur,reUnquctur(quemddmodumficHndd fdrscorreldrijfro^onit) 
tonus.quod eji totum corrtUrium. 

CDiapcntcconfonantiaiminus oftofcmitonijsminoribus continct. 
CNdrrt cu tonus wnus^duo fcmUonid mmord CT «^w c6mdcot\mut\trcs wni cT 
unu femvvniu mimis^eftefemitonid minor d cr trid commatd continebuntM trid 
commdtd fertriceftmdmqmntdmfecundihuius:femiu}77io minore Junt contrd6lio* 
rd. ergo didpente qiue per fr^edentem tribus tonis CT fefnitomo minore conjldt^ 
viimis o^ofcmia>HusminorAuscontinebit.fedquemddmodum fdcxlc monftratii 
ef didpente conjotucntism wmdum oC^.iuum dttingere fcmitoniu minus ^oddudm 
qucdiefimiitdquoquefdcHemonflrdtu effet, edndan conpnanti^onnonumfcpti^ 
mdm dttingaedpotomen, 

Diapencc 



M 24 



12 CDiapentcconfonanria-'ditonojfefquironoque coniungitur. 
CN4m diafenUferfenultimamtribustoms CT fitnitonio mnorc conjidt & 
dimus a-fffqt^itonusjjmultres tonoso'f^^itoniumwinus efficiuntAg^ur dito* 
fitis dtquefefquitvms fariccr ccp4lati:covfondnti4m didpente iungunt.quod in^ 

CConfonanriarum diapcntc & diatcflaronrronus difFercntia eft. Quo 

13 fit V diaccllaron confonantia^adiunfto tonoxonfonantiam diapcntc 
reftifyat. ^ 

^.Differentia hie uocatur ed proporticqud mdior/uperdtminorem. Udm fer eor- \ 

reldrium decima buius-./ubdu^o tono d confondntid diapente, relinquitur confix 

nantid didtejfdrorjfilo igitur tonoiconfondntid didfentCy confondnttd didtejjaron. 

fjl mdxor.ejl iptuY pf r difftnitionemzharmn confondntianmt tonus dijferentid, ^ ' 

correUriimi jUtim ex frofofitione notwm eft. 

14 CBis diatcfl'aron; fefquitono confonanriam diapcntc tranfccndit, 
HDidteffaron enimiper fextam huius^duos tonos dtque femitonium minus contU 

net. ergo bis didte[[dro>2:qudtuor tonos, ^ duojemitcnid tninord continehitMqui \ 

d quatuor tonis cr duobus femitoniis minoribus demvtofefquitonoi Yelinquuntur 

tres torn i^ femitonium mifius,Atuero per dccimdm huius'.confindntid diafente^ 

totidetn tonos cu m jemitonio minore comfleRitur.ago bis diateffdronxfejquitono 

conjbndntidm didfcnte trar^fgreditur trdnfctnditque^quemddmodum frofonitur. I 

^5 CConfonantiac diatciraron,acdiapente,inmaximisfuperparticulan- ' 

bus funtcollocata:* ; 

C Sdm ex diffinitioneiconfondntid didtejfdron in epitritd ffquitertid qu^e fropor ! 

tione collocdtUTy ^ didpente in hemiolid atque fefqudlterd^at tudUfiperparticu* I 

lares :f(]qudlterd crf^fjf^f^^tidfuntmdiores.namdfecwiddO't^tidpdrte.qua \ 

mjxim^e Junt fefe confequentcs fdrtesidenominantur, igitur ha 'confonantid: ex \ 

maximarumfuperpdrticdldrium originib(isdu{lje:in maximisf^perparticularibus \ 

funt coUocAtde^quod eft fropojitmn, \ 

CBis diatcflaron,aut bis diapcntc; nullam confonantiam componerc S 

i^ potcft. I 

€h^c prop onit duds didtefjaron eon fonantiitSyMit duds didpente conjondntidsi \ 

nullat7uon}lare poffe confindntidm.Kdm cr dutcffdron CT" didpente nen in muU \ 

tipliabus :jed.jupcYpurticuldribus funt eonftituti£,0- per primdm petitionem qu<e t 

interualli dd merudUum prcportio eftied quoque eft ^p>ni dd jbnumM perfexf ; 

tarn primi duo ftmilih interualld non multiplicid: neque multipUx neque [upir^ ■ 

fdrticuUre crednt intcruallum. quare neque iUourm fni in multiplui: I 

J Hi], neque :' 



J 



HPi 



M 

neque in fufcrfdrtkularl gaure cxiflunt .cmtius 4idtm confonMid:dut wfu^a* 
vjrnculjTtyUit tn wwlfiflici rationexollocaitddeft.fic fnUSfinantUnomlnt'.hocm 
ioco,vythagoric^mfcqucntci Mdthorimtem/ujcefimus utenimt.aio du^ confix 
fidntU duteffuron mt dudtdidfcntr.nulUmcfficicvt coTtjonamUm. CT «<»» ^^ 
do id um^fift: fed cr qi*otquot conpruntia didteffdron in immcnfim copulate 
twr,crq«<'f^"<'f dupcmCynulUm unqudm confindnodm ex quintd ftimi huius 
iffucrc UdUbwnt, 

CAdiunfto adconfonantiamdupcntctono:nullaparabitiftconfona't7 
tia.itcmncqucad diatcdaron tri(ctflitonio. 



iGlii 1 1 
|bl27|Fl 


1 Numcrorum dria 
i6|Sextamaior 


lT|3384r Numeroru R Sdria 
1' |Rl9ii6lS|s83iL^Scxtamin. 


"|C|5> |D 


8 1 Tonus 


1 |Plx304!Q|i944l ,5clquUon. 


Tai J |B 


iz|Diapcnre 


1 INI 4|0| % 1 Diaccriaro 


i~n 


1 1 


j |L 1 156I F| 2431 Scmic.mi. 


i i 


1 I 


1 |H| 5> Rt ^1 lonus- 



Ctf /I \7ufin0rum congrejfus rtcndum conjondtttiafit'€Uj^nuf9ftdmcn muftcire^ 
futdttt mdo,modi*Umimbuf<iued^rum.fextdm<iuc:quotlcx imfledtur uodbus.no^ 
jin nuncHuncupdntj^ quatuor tonisdtqueumddicft,hQcefl fcmitomo minoreico 
fidt. qui quod nondHmconjonamidfttfdtet.A.cdfioenmdO' ^ tcrndrmma'bi 
ndjLm: mimmos faUcct numeros confindnM dupm^.^TC d wmcndrium dtqut 
od^ndTitm mmimos mmmstonuCT d**^o ciHdCTUcmdt e feptcm (uprd uigtn- 
tt^vd inker itemdt [decern cr Jfx.pa tertUm^uinti dxithmctices e ad fcon* 
tincc ^fqualtfTHfn ^fefquioCiMmmiqudre e dd fcontinet didpentedtque ddiun* 
tbmtamm dt mdnifcjiiwt ejieadfx jfdUctt dd 16 non e\fe muldplex, nam 
feptcm cr uipntimon continent bis out tertiodutdeincepsfedecim.nequeefuper 
ydrticuUris ejl dd f.ndtn drU nutneri eddfeji imdecim qui numerus jumm^ 1 6 
cdri nu\U eft.trdnjcendit enm undendrius fedaidni dimdiuvt. i^ur ddiun^us 
ad confinditum didpente tom4S\nulldm pdrit confondntidtn cr jimili dr^munto 
adicao dd confindntidm didtefjdronfefquitono fudU pt confindntid ut ex fecm* 
ddpgurationeperfddlepdterepotejl.jittdmen euphond uocum coTtgreffio: qudtn 
Item fextdjmnuncupdHt{ed que dprimdcontrd^ior totd dtjlet dpotome, eji ergo 
htc minofiHU uero mjm.conlldt enim primd ut di(lwm idtn eftyqudtuor toniiO' 
WfU SefwfecHiMdduao trihus tonisO'iudhus diefihus .Vrimdmfindt pdfhypdU 
lnpj:onddmrCcm:fccundamucroqu^comrdaiorcj},fondt hypdt^bypdtoudd U 



cbdH on 



1 



CimtMr:fcquens lierdecUrdbit. 

^^ CQuo pafto diapafoDCPp fonantiarin chorda fit adiungcnda. 
1 A I B L _^ \ _ Confona ntiTdiapafoni 

fLn^ec c onfondntiarum ut in Btr^ frohlematum tefl4tus ejl AriflotdeUelegdntiff 
tjfd fukhfrrimdqueefl.chcfdam ergo d b/ecoffr medium fernomm c.O" ^f*id ^ 
bdd cbeft dufld tnterualh habitude: ergofer diffmitionem ab ddcb conpnat 
didfajon. ^ 

jp CConfonantia diapa(bn:in fcx tonis minimc confiftit.fcd quinque sm 
pliorricx vcro roni$,confonat conrra<5tior. 

CKdm per ukeftmam fnmi,(juinque'coniu/9t6iifecquio6lM4i:mmus duplici inter* 
UdUocQmungmet^CT fer uic(fimdmprimdmeiufdem:pxconiun^i mdioresunodu 
plici interudUo euddunt.ergo qmnquetoni,minores [untdid^dfbn confonantid: CT 
fiXyCddemfunt dmpliores.confindt (rgo didfajdn quinque tonis dMfliorifid CT /f x 
eddetn moduldbiturififerior, 

zo CDiapafon:ex diatcflaron & diapentc confonantis coniungitur . 

CSdm fer decimdmquinmm huiustdidteffaron O'didpentein mdximis fu^erfdr" 
ticuUribus fint coUocatoeMper umdecimam frimiiduflex interudUum ex duobus 
ntdximis fuferPdrticuldTibusconhc^tur* CTjit^plex interudUt4m:conj6nantia di 
dpdfin intaudUum exijhtigitur confondntidm didpdfin: didteffaron 0" didpente 
conpndntut fimul coniungunt^quod eft pr&poftttim. 

CConfonantia diapa(on;quinquc tonis & duobus femitoniis minori- 
ii bus,qu2e tonum minimc compIcnt,perficitur.Vndc quoquc manifeftii 
cflc potcfticonfonantiam diapafon folo a fex tonis commatc diftarc. 

ilPfr pracedentem enm didtejfaron tr didpente con^ndyitldfndidpdfon iungtit^ 
didtefjaronditteper pxtdmhuius duobus t^nis^ (emitoniomxnoreconjtdfemon-. 
ftrdtn efticr didpente per decimdm tribf/s tonis pmitontoqut minore, dt duo torn 
(^fermtoniu minus jO* tres toni cr itidem pmitonium minus fimul confldtiiquin* 
q^cfficiuntur toni dtqui duopmitonid minor d tonumnon prrpdu/nt: uerwrnab co 
deficiunt commdte.iptur confonantid didpdpniquinque tonis o* duobus pmito^ 
Tths min0ribusyqua tonum implentyquemddmodum idmpropofitumeflyperftcitur 
CorreldHium ex demonflrdtionis cdce notum effe potefl.^x quo liquet perfdcile ef- 
p in neruo mifficum commd perueftigdre.fi dm in eo <t prindpio conftitutis ^conti* 
nudtifquepx tonis, ^ db eodem nerui initio ddmtdidm chordae notdm intenp did 
pdpn conpndntidiquod inter medidm nerui notdm,^ ultimum px tonorUmp* 
gnum contineturyex pritpntis correUrio eritcommdtis interftitium. 

c Dcm# 



r 



M 

CDcmpra ex diapafon confonantia diapentcrrclmquirur diateflaron.^^ 
U ex eadem dempra confonantia diatcflaron.Tclinquitur diapente.dc 
ptis autctii ex ca diapcntc & tono.relinquitur fcfquironuJ. 
CPnrrtjpjr^ C fie u^da flat m par ftmdtirnam cognM jt0iUtcm CT ^cr fr*f. 
ccdfntem.Sdtn fcr,^xcedcntcmdiaf*afon quinquetonucrduobus (imitoSus mn 
rtoribus conftar.d qHtbus ft trfs fPTWiCT u}U4m fcmirmmw mitui^ wUas,que per de 
amjmt huius dufcnu con[on4ntiam (ontincnt.nliqkumturiuo wni CT fimironm 
vv.mis.qux fcrlextzim huius duxteffaron confifuintumcfficitrntMn^ta igitur ex^ 
dupafon corjfonAnti4dUfcnte:relinquitur diatcfjarotiyquod (flfftimum sccumdu 
fudtw fadludtc dcclsratur.'Sam ex quinquc tifnis cr duohus /€mi:vmis fpinori- 
bus :fi duos tonos ^ ftmitonium minus Mas. rdinquuntur tres ton'iO' fi^^^ 
mum minus. Jnttuvt con ftmiliter.dcmftis tnm d conjcnantu diapapn^hoc ella 
qutnquctonis CiXdi^obus {emitoniis minoribus'Jtmftis inquam quaiuor tonls CT 
fimitoniomtntn-iireUquus efl tonus mus CT femitonium minus. Ltquotdiefis, 
quot dpotomds yquot dcniquccommdtd contineat diafajdn: deprehcftfionis funt 
f.iailifrut.in nulU tnmen ^qudlitnitotn iffd diuidua tfl. quandoquidtm dupafbn 
m nuilttflicirjtionc conftflit': quaommnotn quotlibct Jquds fro^rtiones qu^e 
vmlniiiccs nonjmt,fcr/ixdg(jimdm noni arithmcticcs dwidi non fotiji. 

CNuIIafimplcxconfonantiarinduoxqualia^ccrro, conftitutoquc nu- ; 

mcro diuifibilis eft. 

€i$unplicef confindntids uocdtms didteffatonydiapentc, diafafin.de didtefjaron 
Macma-didpcntecoi^fldtiqUiteX fitpcrpdrticuLmbus intnudUisfur^unty qu^e 
fer quinmrn pftmi nulla pdt\o hue in modum dirimi poffumtfde duxpafbn uno con 
findntUudcm fiibit iudiciuvuSdm quonidmmimmieius numcri funt duo CT W^w 
cr duoquddfdtus noncPrJ^ttur per c<mreUrium triccfim^fecwnd^ fiiumdi huius 
conf>nd7Ui4 diapdfin qu^ confiflit in prvportwm duomm dd wnum, minimc m 
duo ^qudlidpdTtutur,^ codcmquoqnc iure mque C4dcm ccnfindntia in plura 
duobus dimctictur,dinm€turqu€ ^qudlid.lt profc&o ueUe hoc pd{}o confinanti* 
^m didpdfinin plutd atquddiduccrtiefl in gcomttricis didmctru?n coflc quddrdti 
uclle commenfuYdn,fid id ultimum:mufcum non e(f. 

CDupafon ac d iareiTaronrconfonaatiam non'effe. 

| E 8 I t ? I g 1 I 

I . 1 I I 4 I ^ I 1 

1^ A b I C I d I I 

Ctt/im iidpdfin dc didtt(fxron fit dudxum uocu^duUis ydmendquecongrcjfio, 
ut quemddmodum compulptyr pfj*^itVTtus:non icchcotdmeneuejutdidpdfin dc 

d\4i 



15 



III 16 

diateffdroH confindntUm dicintneri.Tdmetfi Ptolomaopcus qudm vythdioricU 
hdc in re mfumfttiquoi monflrdturfadlUmum,efl.S'mt cmm dO'b binarius Z7 "^ 
nitds^minimfcilkct numeri conjonantit/e diafdfdn.O' c dqudtuor cr trid minU 
mi idemtidfm canfindfitUe didtefTaronMco cinaCT uenUt o^lonarius qui fit e: 
Cr din bo* ffcnict temdrius^qm fitf. fer tcrtidtn quinti drithmctices,e ad fcon ^ 
tinetdufUm cr fifqf^i^^rtiamtquare diapafon ac duttcjfdron.fed e o^ondrius,nS 
eft multiflex iid f temdriiimyntque fipcrparticuUrisiquod ewm his continent CT 
injuperbinariufnqui temarij nanfdrs uUdeft fed partes, eft cnim o£iondrius ad 
temdrium'.&plex fiiferbipdrtiens.Hon eft igitur diafdfin dc didteffdron confix 
nantid'Omnis enim conlondntid*d44tin/uperpdrticuUrt,aatmmultipUcigen€re^ex 
diffinitione cofiftit.tt in hoc facile cognofd poteft ex nono problemdtum libro quod 
Vythdgoricis conftnfit Ariftoteles:cu/m inquirit cur non Us diapmte^ dut bis did* 
tefftrpnreddi confinantid poteftyUtbis didpdpncodptdrijolet.Hocdnquit^ideo eft 
quod didpente confinantid pofitd in proportione pfqudlttra eftzdidteffdron ueroin 
ftfquitertid,quodfi duo fefqudtcri out fifquitertij numni ordine difponanturiex ^ 
tremiruiUdnt inuicem proportionem hdbcbt^tt.neque enm/uperpdrticulares m* 
que nmltipUces effe potertmt,dt di^pjpn concinentia quonia in dupUri proportio 
ne confiftit:hoc gemindmyquddruplam inuicem extremi tenebH^t.habebuntque ^ 
portipnetn »y ides ergo quo pd^o AriftoPeles confinantiarwH proportiones folds fi4 
perpdrticuUres didt multipUces efficit:fuperpdrti€ntes:qudfiprorfusnull<t fint^re* 
pudians^itreuera Vtolotn^i cu/tn Vythdgonds ntdgis in nomine quam in re ipfa 
diffenfio putdndd eft.fed de his hd6lenus, 

CDiapafonacdiapcnte; in triplici confiftit ratione.eflque diapafon 
ac diapentc:confonantia vna. 

COjiJd enim diapajon ac didpente in tripUrdtione confiftdt:hoc ideo efl^quodex 
duodccimd primiex dupJitidtque /efqudltcrointerudUo triplex ndfciturinterudl- 
lumMupkx d44tem cr fifqudltcrum: funt confondntiarum didpdfon CT diapcnte 
iytterualldjgiturit0td4£confonanti<e did^fon dcdidpentezin tripld rdtione con^^ 
fiflunt.pd cum fenfu iam fdtis exploratwm hu/nc coricentum moduldt^ie: fuduitcr^ 
qucdd .uiditum peruenireiergo per diffinjtionem is concentus cofdndntid eft^quod 
eft tot urn propofjtu^, 
CDiapafon diapente actonuy:mcloscitra confonantiam cliciunr. 



16 i 


i7 




1 


8 




1 


1 


e 




1 


t 
1 c d 




1 


1 


-p^ 


1 


1 I 


1 


1 


a 


^ 


1 i> 


. 1 












G ii. 


Mc/oi 



^ 



J 



M 

CMfloi hie medtms piMiemdi<rilHts accemm u^uUtionem,4mmmqiteflkri4^ 
tnu uQCwm congrefjumyfidqu^d iidfafon dUpenteO' tonus firml 'mtihymdos co 
J};tuf0it.[iatm notwm ffi,fiam h^cfonorum uoculatxci^duitetMt ^x^^mentidiU 
fdtuf,M47thusdcddit.pd quod €onfQnantiamm4lLm foment: cftenditur, qucniam 
cnim f>erfritccdnttemduifjlonacdi4f>cntc in trifU proportienc c^nftjHt.fmt rr 
godb f 714 cr U4twm minimi coti^ridnti^ diaPdjon dc did^ntcr.CT c dnoucm CT 
c{\o ^minimi m4mmtoni.duc^ cinaCrd inth cTW^ww^f c f, 17 /H/rcff CT 8. 
inter qua ejl didfdfin diapcntc atquc tom$.fid 1 4d fnequcjuperf^ticukris vc .^ 
queniultiplcx:quimmoccontinetftcY Oltrescius o{iMids/j\qu€c dd^triflusfu ' 
f erf aniens o{ljUiis,non concinit i^nur e falkjua con finantid. quod toium ejl 
froj^ofitunt. 

CBis diapafon confonanriarinquadrupUriconftiturarcp^titurhabitu 17 
dine. 

Cc^u en'm bh d'ufafin in quddrufUri confijldt: jldt'tm ejl manifcjlutn^ Sdm 
Tcr decimamtcrtidm primuduo duplida intaualUiquadruflcx itot^untinteruJ- 
lumJupafin d^tem in duplm coftftfiit.i^iturbis didfdp>n quddrufUrem mngit 
habitudtnerH:qua efl n^ultifkx.CT' cww bis didfdjon ad duditum/uduisy emodu- 
y Utdque prrwrwwf.wtid quoquefrnfuptis ferceftum eji.crgofcr difftnitioncm e- 

uteonfonantid,quod adtmonflrdndum. P>ffwgorin CT prions mufici omnes, 
conccrttuummoduminterminisquddrufk dtquc infinibusconfondnMbisdidfd^ 
fon foftnnxrruntinon tetnerc lonpus fro^tfjiydut quod interliUos terminoi uni 
cuique fd6\us 4 ndtuTdrepnitur Jua uocis mciduSydut quod Jhndulus illecdnoriU 
lis (ut iam quoque di^um eft)uijus eftyquoduefeiidm idm relinquunt mediocritd' 
tem,dut quodhdClcnus contemflatio'fatii iffis efji uifd cjlddmuficam injhtutio* 
ncm.f^Jleritds M<temdd terdidfdfon uel^' dm^lius ddduxit dd tenmnos ufque 
^^uf'lelongwseudgdtdidequibus ncceffdridfpeckldtio non incumbere uidetur, 
fed fducd fduds Rrinxiffe fttis rrit.'Sdm qui moduldtioncmfuprd bis didpdfcn co 
^nofcert defydrtdlbuntifdcui danonjhdtionc ut fracedentid ]^erdf\cnt.bis enm di 
dfafondcdidtefJarontinfnrofortionequincufUfefquitertid conjijht, ^ po'indc 
fldnt confondntid futdndd non ejl. bis didpdpm dc didfente in f^roportione [efcu- 
fUiCT Wf^ confondntiis dnnumcrdtdjcrucro didfdfoniin poportione odufld. 



COmqcs confonantias:in data chorda fuoordinc fubiungcrc: 5c cas 
fcnfupcrccptibiliter cxperiri. 

'|A C\ d\ C\ t| gl h T 

5lt 



z3 



1^ 



Csit d hidm cf)wJ4 in qua pro fo fit urn fit ccmpnarUs iiateffaron, iid^tnte, dia- 
fafonydidpafin 4C dUfcnte,Cr bis diapafin fttuare.coUoco infiptoa brachium cir 
ctni mmobU€Cr<^totius chordc quartnm partem extcndo circini brdchiummobi^ 
U:CT w termino cius pono notnmcjeinde extedo idem brachium ad eiufdem chor* 
de partem tcrtiam : CT w termmipono d. mox ad wtius chorde partem mediam 
quam deftgno nota e. deindeeodem bracbiocapio totius chorde bi[fenJ)oc cjl duas 
tcrtids :CT in termino biffes pono f. mox extendo circinum ad chorde dodrantem, j 

hocefl adtreseius completasquaruts: in cuius fine affigo notam g,tumc fic a b cr 1 

c hy^erquarmm humsxconfonat diateffaron.d b O' ibper o£hiuam:Jdapente, a b \ 

Cr^bper decmamo6htuam:diafdfin.a bcrfbfer uicefimamquinmmididpafin I 

ac didpente. ^ftremo uero abo^gyf^ pr^edentemibis diapafon. suppone igz* I 

tur nmfxcde hemffherimt(enfm fmgulis chorde notis , ^finos ad totius chorde [ 

mm iiiii^enter attende: cr juo ordine propofitns condnentias annombis . quod j 

promptius exferiri ualebis :fi chorde a If chordam equiJondm,unifin<imque etiam ! 

coUocaueris , cuius (onum cufn fmgulis [edionurndb percuffionibus non figni* 
ter dttenderis* 

jpCConfonantiarumhocpaftodigeftaram : finis confonantiac diatcfla* 
ron,ad finem diapente lonat tonum.ad fincm diapafon : confonat dia- 
pcntc. ad finem diapafon ac diapente: inconfonus . ad fincm vcro bis 
diapafonrconfonac diapente ac diapafon. 



M g I "bj 



Csiwt d^Cydyeyf^gyb modo qui diCius cjl digejlce conpnanti^idico c bfinem^conp-^ 
nanti^ duteffaron ad d bjonare tonwrtiy adebejfe diapente y adfb inconfonum 
^fUyddgb uero confonare didpente ac diapafon. Ham ptr pr^cedentem db cr c b 
ej\ diateffaron:^ ab^db diapenteJeritpta ergo abcTcb diatejfaron covfinan 
tia^dbc btrdb diapente: per correlarium decirmt huius relinquitur tonus, quod 
dutem relinquitur eftcb ^d bugiturc bad d b finat tonmn, er quoniam per 
pr^cedcntcm ab^^eb concinitymodulaturque diapafon-, fubjlrada iptur a b CT 
c b diatejfaron abdb ^ e byper uicefimamfictmdam huius quod relinquitur cjl 
dijpente.atqui quod relinquitur efl cb^e b:\gitur cbadeb conjbnat diapente. 
cr quia per vr^cedentem abo'fb confonat diapafin ac diapente: fubdu6kt fgi-^ 
turdb^e btonfondntid diapafon : quod relinquitur ejl diapente,quod autem rc- 
linquitur eji eb^fb. iff tur ebcrfb corjfonantia ejl diapetite. fed permodo 
honjhdturn ch^eb etiam diapente ejl:igitnrc bcrfbejl bis diapente. at per 

Ciij deciynam^ 



M 

i^ cmdmfextnm huius his Jufente confcnuntld cmnfiminon fotejl : i^turc b dd 
fbinconfortus efl^pojhnnocjuonum utex^rdUticntin^tHm (fi abo'gbconfb' 
TJat bis didpdpfi.demftn inturab CT ebdidpafon confinantidirdbiquitur ( b O* 
g b fffediafdfon,dt<juicb^T ^b fcrficwndam fortanhuiustnoflratacfteffedid* 
fcntf. iglruT di\wn6h cbCT^b cottfonantid iidfente ddcb o* $ b,ccnflituu 
tur dia^etjtf dc du^dfon. carffnJtn^o c bcS^ didi^trtte dc iidfdfbn. quod ejl 
totunifrof*oftt4f?i. 

CSic pofiris confonantiis -.finis diapcntcad fincmdiapafdfi modula* ^o 
tur diatcfljron . & ad fincm diapafon ac diapcntc: modulrf^r diapa; 
Ton. ad finem vcrobis diapafon, cuphonuj eft: fed qui nondum con- 
fonanria eft. 


^A cl d| c| i\ gl^- bl 


Cej?o fff^ccdcnt'uon hyp0thrfts:dico d b fini didfentfydd cb confcndTc didteffd- 
ron.dd f bidiai'dfofi.cr dd g.b fcHdre dUteffdron ac didpafon^Sa per perwltima d 
b^ tb cSfoHdHtu (ft did^dfon^O" dbcT db didf(fttep^bflrd^ iptur db^db 
dLip(nt(ydbdb CTc t confotutntu diapjfon: frr uiceftmdmficii^ddm huius rr- 
J^ , linquitUT didtcffaron.quod dutan reBnquitur (ft dbcT ^ btigitur dbddeb con^ 
fonjt didUffdfon, CT q^^ ^ fr^tcednHi vtovfttdtum eft (b^ fb€[f( dwpcw- 
tCfCy ni0icd b cr fb eff( didte[fdron:(rgo p(T ukefimdm huius db ^f b ex iU 
lis dudbus conflan,codlftdqu( confofidntidymodulabitur didpafon, Kurfunt cum e 
hcT gbin priec(dcnti moftflrdmftt didpafon^CT eb CT fb didp(nt(:(rgo f(r ui^- 
ccfimdmftcunddrn huius f bcT gbrft didtejjdrorj. cj-db crfb numc monftrdf' 
f3 (ft didpdfbrtJgitur dbo' S ^^ft dupdfon kc didteffdron. qu<e cum moduU* 
no fit (uphond , Crf-ttw monfttdumt uicefimaqudrtn huius non (ff( confonan-^ 
tidtn: tutum liqu(t monftrdtum propofttum. ^ (x hdc quoque pd7it(r cognitum 
(ftfindn dupdfcniddjiiumdidfdfon dcdidperite moduLrididpente ut (tudfb. 
^ ddfinnn bis didpdfon:c^onaTe didpdfon, 

CConfonantiarum ficcollocatarum; rotius chordcatquecuiufquefc* 3^ 
^ionis nuracros defignare. 


1 2,4 1 i8 ( 16 1 12 1 8 1 6 1 1 


1 3 1 c 1 d ( e 1 f 1 g 1 b( 


Duco 

\ 



in 28 

Cduco duoytridr^ qudtuorin^inukan:^ tumerum ind^ furgentem dtque fro* 
Ju^umqui habeUt jicuvddm tertuim cr quarmm , powo totius Uncamtmerum 
quern iccirco uoco numerum a. b,ab quo demo quartnm partem o* reliquus fit c. 
t.'Cr eritfrimus numerus dd iptitmfefquitntius , quare dUtefpiron. ^T ^t eo* 
dem numero dimo partem tertiMn er reftduusfit d hieritqut a bad bfefqualteXy 
qudre confonantia didfente.CT iterum ab db diduco partem medldtny^ rtfiduus 
fit e b: rritque db ddeb duplus. quodrai didpafon confonantia inter eos exur* 
get,0' numjfi d b/umo foUm tertidm qua fit fb : erit ergod bddfb hdbitu* 
do tripld. continebu'nt igitur d bo'f^ didpdfon dc didpente, Kurfiim rtwne- 
rid b fold qudrta fit gb : erit a b dd gb quddruplus, quare dbcTg t^ fr^^ 
fuimeri bis diapdfotj,f4nt imque wtius chorde dbo' cuiufque fi£iionis eius ficim* 
dum dffigndtns confoTidntias defignati itumeri^quoderdtdemonjhrdndum, 

CQuorcunque harmonicas medictatcs afsignarcrintcr quarum termi- 
3^ nos cordmquc difFcrentias,omnesmuficae reperianturconfonanda?. 



1 Harmonicx 


iz\8\ 6\ 


lEpicritus (be DiatefTaron | 


jMcdietaccs 


6 4 


1 3l 


Hemiolius (ab Diapcntc | 


1 1 a lb 


c 1 


|Duplaris |ac Diapaloa | 


IDiflTercnriac 


1 1 


I 1 


( Triplaris | c e Diapafon diapcte 1 


r 1 


M 


c 


i Quadruplaris | b e | Bis diapafon. | 



CHdrmo^mca medietas in drithmeticis diffinitn efl: quandotriumterminorum ut 
mdximus dd mimvmm^ita differentid maiorum dd differentiam minorum. sit e r- 
go c quicumque fmrncrus partem tertidm hJbem quae fit e. duplo cifitque dupld- 
tus d^manifeftum eft dad c ejfe duplum.CT q^^<^ ^ continet trid e:ipfum a continct 
fex e.dddo eddc crfi^t t CT ent notum b dd c effe fifquitertium , cr h continerc 
qudtuore^dtqueeeffe differentid' b dd cAtidem quid b cotinet qudtuor e^O'U^on* 
tinetfixxid erit fifqualter dd t.CT^wM |> ^equatur qudtuor e^o^ d fexiergo differe 
tid d jSfv dequdtur duobus e^quaefit d^quia cnm d dcqudtur duobus e: ago d <J«- 
lus (ft dd e.dico ergo db c ddtaefje hdrmonicdm mediemte: inter cuius terminos d 
c cr eorum differentids d r , omnes mufica confonantide reperiuntur. ndm dddb 
mdximi dd minimum monftrdta fft proportio effe dupld : ^fimiliter hdbitudo 
ddde differentiae fcdicet mdiorum dd differentidtn minorum etidm oftenfk dupld. 
funt iptur per diffinitionem dbc termini in harmoniat mediemte conftituti. At fi 
b ddc com\rdres:monftrdtus eft fefquitertius^qudreconfona^tidedidteffdron inter- 
udllum^crft ^ ^i k'-fftonfttdtus eft fefqualter dtque hemioUusyO- didpente inter- 
ujlum.c fiaai cduplii hdbes Cr ccfondntidm diapafon.fi uero cade contukri- 

G iiij habej: 



I 



I 

J 



M 

^^bcs rripfww, cr f ^ ukffimamqubimm hulus ccnfondntUm cnn^ofitntn iU^4i 
[en ac dufcnte. cr ji t ad c: quadrufUm, CT f^r uiccftnuimpftiTnam huius bis 
diipjfon.at cum cetera monftr^trnftut non ejicconfcnantius : conjiat diatejfaron^ 
dtafcnu^duifdfon.diaffafonac dUpcntej^ Usdiafdfon omnes c^nprnantids qui* 
dui in ihifUmsfe <xaccre folcnt rmficiyintertnminos harmonice medicmtis CT 
bcrum diffcrcmias, fuifft reform, CT ft duxeris binArmm in dbc^in illos qui 
cnde^QUcncrintyO'q**^^^^^^^^^ ^" frouenicntcsiexftftmaficiiTidiarithme' 
uca cc^fiofcac fromftum c(l tuies conjHtui harmomuun mcdUtntcrg^ conftmiles 
tntcrjuos tmfMnoscrfr<^i*f^* tcrnunoTum differcntUst confinantUsmufiats fir^ 
iudntan.ix ^<^ fucntft qucmaiTiqu^dltrrum^numcfum in illos ttrmnos duxe^ 
Vi,pUcmt mtficn diuo %eucrifiQ quadra^ ftno o£lmo ca^itt fccumdi fu^ arithmu 
rices ad omnes confonantuis muficas comfie^cnddsiduss ordtnarc mcdietzitcs hdr-^ 
moniaiiyunAm in iiwplrfn, CT aitctdm in trtj^idri:fed CT idem etian^cri lo[fe Ufu 
fold ionjluuraAam jdtis monjiratum arbitrAPmr. 

CQuorlibct maximaj harmonias,quarum qurlibccprimord.um confo 35 
naDtiarum,confonantii(quc contincat omncs.conftirucrc 



|Maximabar.|i4 |i^|i^h^l_l I I 1 

jMaxima bar.| ix| 5> I ^ I ^1 | Epopdous |bc Tonus [ 

I I a I b I c I d I I fcpitritus | a b | Oiarcilaron | 

i Differcimat. |_3_l_l \ I |Hcmioius_|aj| Diapcnte | 

'1 I c I f j I Dupl ans j a d | Diapal on | 

i Diffcrentix/ 1 4]T^ i "1 Tnplaris ) dhlDiapa(odiapcte.| 

I I g I h I I Quadruplus | c h | Bis diapalon. | ' 

Cm jxim^m hdrmotiUm uoatnt : qudndo quatuor folidorum tnminoTumJn ^f o- 
mctnai medi^mte conjlitutorum , inter maximum , unum mediorum ^^mtum 
medieas ^rifbrnmai contiuctur,cr rurfum inter mdximum tcrnUnorum, dlterum 
mediorum, cT Tmmnmm con^eturharmomcsL l^ediet:is geomctricu: efi quando 
tcrmnorum^Jl froportionumftmiUtudo* Arithmetic uero: quMdo terrmnorum 
ej} dtfferentucrum ^equahtns.quid harmonUa iam di^um eft.soliditer/mni dicun" 
turtqui ex trium in ji Uterum du^u froducuntur. fed hac omnid ex drithmeticis 
qiMm koti[fimd funt. nifnordium confondntiarumidpfelUnms wnum. Cdpio ergo 
d mmerum quemcunque qui ficwnddm cr tertidm habedttfitque eius/icuni^ /. 
Cr tcrtid b. duflo dcfi^^^f^^^ **• quiquiiem 4 duplus erit dd d,dddo h ad d 

fitque 



Hi ip 

fique comfofitus c.mtque pfyt^ertUis di i.fei & cum a tnciiftratusjit iuflus 
ad d:erfff per undecmam fvim buius^ dadcefi Mtualtcr. Vr^tttta ffhmmm 
fort^m iiMMffi itXT comf^fitns fit kcmum pibaddtffc fifqfMkerHm.qu4* 
rcf€Y eiatdtmwtdecmam fttnui a ad befifefqukertius* Ab a c igUur quimon* 
ftratusefififqualter^huo ab fifqitkertioifn qmnmmdtcmam ftimiy relinquu 
turbad cfifquioituuus^it fr^Urea gdiffercntid dad c^ma d coioinct tna. b:c 
contm€bit,qiMtuor,o*d/bc.ergogdiffcrcntUd ad acominctduokiftimqueg ad 
b duflus^j^qmafeji mtdiemsd^^befififqualtarmnaddiergp bconi'mtttriaf. 
^ imqueftmid fars b.fid CT dfifquitertms ad b ad^fufir mm tertiam far-^ 
tiifftusb: er^odiffnitia dod b q^fit e aquaturf.dk0cr^oayb3Cjdmaxmam con 
JHtucrc barmonidiqitonu conjonantiaru clementUy^ omue cofUiliturcSfonatiam 
liaaadb mSftratusJifquitertius.O' fttniliter c ad d jefquitertius cStinet igitur a 
ad b ccr^d d per difpnitioneigcometriatm mcdktati.^ differentia aadb maximi 
ad «»M mediQYUfn ejl Cy^r badd eiufdem medtj ad mmmtm efl f:^ e erf won* 
ftratejimtaquan.igttur fadiffipitiomm aadbcrbadd conjhtuuntur in arith^ 
mttiatmcdietnte.fedcr^dddmaxmus adminmrnn monfiratusejiduflus: CT 
/imtlitcrg differentia a ad cmaxmi adreliqimm mediorUfiemo^jlramduflaad h 
differentiamc add eiufdem medijad minmtmtjgitur fer diffimtionem acd confi 
Jiwnfin harmoma medietate, Conjlaiiptur ferdiffinitibnem a,b,c,dfi foUdijint 
conjHtuere inaxmam harmoniam^quod jifilidi non ftntiduc quemcwnque mmc* 
rum in queni^bet iffirum , CT frouenientlolidi in eifdem babitudinibus quorum 
cuiuflibftUUerderumtunims, rmmerus in illos du£lus 0'fi>fS^li eorum adftngu* 
los^quicoxfiabit maximatn conJHtutam ejfe harmoniamftd iam oftenfus ejl b ad 
c fifquio£biUHsnpturPer diffimtionem bade continettonum confinantiarum fri* 
mordium. CT rf 44 fc eft jefquitertius : igitur aadb continet diatejfaTon^ ^aadc 
monRratus eji fifqualterii^itur a ad c continet diafenu.& d^dd duflus ; iptur 
a add diapdjon. O'dadb triflusiigitur dadh diapafon ac diafente. O'^^d h 
monjhatus quadruflus : igitur cad h continet bisdiapafin. O' quoties duxeris 
quemcfil^ue immerum in ajbyc,d aut in produ6ios ex illis: adiuuante pptma /?< 
cundi arithmetices cr diffinitionibusytoties conjlitues maximam barmoTHam, to^ 
num o* omnes confonantias continentem, ejl igitur effe6i urn quod froponebatur* 
tt ex hacintelligcrepotesiejfedemonjlratum qu^ diuus Seuerinus affert debar mo 
nia cubi quddrageftmonono y^-de maxima harmonia quinquagefvnoquartocapi 
tef€cundifu<e arithmetices, 
N 
34. COmnis numcrps ternaria progrcfsionc ad fe adie<^us: omncm con< 
fonanriani in arichmctica mcdietate complcr. 

H covfonantice 



1 



1 coirfBninM in ofi thmrttcd meduUtc 



— \lir^u!jiter.JiUpente | \bemol. 






[}\ 4. 



usAu 









A (€ct4vdd ifwrtafc 



A ftrru umiuite 



io_2^ o^l 40J loojiooj 300|400_J 



di^ 












1^;^ 



pd. 



Bi5 jjupafin I BiJ dufafrn f 



JupfWfg 



I 



tLlnteUr^itur tmiana pro^ejpcvc ad fe dddit rMmerus: qudndo additur frinto 
ad ft ftmtlyidndc his ,dtindc tcr-dico erio qucncu^que numcrum ad fi hoc paOo 
additum:omncm cojlituere cort/onantum mufxcam. Sam (imti ftbi adie^ur. <f«« 
jicx ad fe tffidt^ diafdfiu ccm/onantiam.c hii ftbi addiWS'.ad frimam additio 
nem frfqualtnumconjlituit cr diofcftte ,a- ^d ft triflnm atque dta^ajcn diafcns 
tf.nam twprwwdddrrwwf ttircr in ficunda tn continftur, tcrnerofibi additus: 
ad/icwtdam addttwncm facitfefquitfrtium ^ diaUfpnr^htMm frhnus nurmrus 
hie qu4ter,illic ucre trr coniinctuf.cr ad fe qHodfUflum facit cr bis diapafdn. in 
f>4Cijiriirprojrf/Jro»c ownis contineturcoTifomntiamu/ka.O' ^w continue m 
merer itmtx eius additione/ut^entiumyipfe numerus addxius efi dijferenrut ler^o 
reperteconfinanti^ in arithntetica medueate continentur.CT fx hoc cognofcitur, - 
cur in omnium umtatum qudtcmaria progreffieTtt qu^t ubique denary Vythagori* 
ci fUnitudtnem impkt:cmncj conjonanti^e mufice^CT in aritkntnica qmdem wr* 
iutatt Tcferiantur* 

CTonum 8>c omncm confonartiam fimpliccm:in duo arqua partiri,vc 3S 
raquc mcdicratum pun(^ ^n chorda, geomcrricc monftrarc. 
H^r^cdetes /cftima pcudiyO' uiceftmatertiatertij: ptiftaidunttonu^diateffirS 
diapente 4c diapafin in duo^ua diuidinonfofje. h^cuero monjlrat quo va{io 
td omnia foffmt in duo ^tquapartrri.ncchoc quidcm nfu^t, Kam fticccaentcs 
contcnduntideffidncnpofjc arithmetics certo,coytj]itutOijue numerOjatque rati* 
onali hdbitudincih^c uno Id eQiei pofje gcomctnde fine muncri ccrtdj conftantiqi 
fdtiQne^ 

Sit 




tLsit ergo i4td cyrid d h fupm4rin,<]ud iuhedmur integmm pmUpnlmn^tr cS* 
finintunmm du^ffdron, didfcntCi dc dUPdfin uerd mtdld fefenre.fddo 4 here 
btonum.db & it> ^i^telfdrcn.d b tT ebdidpentcdb crfb^j^n: to. qui in 
fracedentAus mtmfhrdtusfjl mcdo.dmde in inftrim linad d^c in definite qudnt 
titdtisttdpio d b ^qudlan hu^ fit f axon d b^iO* ^ ^ continue aaudlem line^ bcfi 
pfnori-cr d vtmSlob unfits czcdPtotinaedm bd ^quaUm chords bd, ^bta- 
qualm chcrd4eb€yf!;rbfchordcebf,cri»teUigoqudtuordintidios circulosd c,dd 
d eydf,CTdfun3cb educo fcrpendicuUrem line^ dcdd circunferentids fimicir 
culorunidCyd d,d tydfc fun^ldubicos contin^t lin^d fint %,hfi, k:ddqu4t 
funCid educo dg^c g,d f7,d byd f,e i,d kJ k.dt per nondmfixti geomctrieid bddb 
gyUt bf ddbcfdcioigitttr inchordd ^perioYCdb\\n<tdm bg ^qudlon inferiori 
\ine<t bg-cr cum proportio dbddgb utgbddc^but fraoflenfitm ejii fequi" 
turtorutrndbcrcbeffcinduo <equd dimfitniyO' fu/n6lt*m g effe Medium ueri ft 
mtonij fignum,^ fer edndem qua ftoportio dbddbhed fitb hddb dy/eflo fgi* 
turhb in chordd d b^Udlts Une^b b.prr idem ut prius:ed eritfrcj^rtto db dd 
h h qu<£ hbddd b.quared bo*db didteffaronun duo ^equd'f 4rtxtdefl.tr todem 
^aC\o fji(\d chorid i b (lipf nori oequdli linea biy^ chordd k b aqudli line^t b k: 
fnonflrabis ^nfondntids didpentey^' diapafonin duo aqudeffefdrtitds.O'reue^. 
Yd hoc ^a{Io ue^d fchifmdtd dtque didfchi'^mdtd , ciT diejes tetdrt/tmorids y 
dtque rcpcrids d[ft^natis commdtis : diefcos , atque compUti huius femitO" 

H ij nil 



M 

nii\nteraMs*.^fumftU (ut in frr^ccdtntibus fd£iumeji) mcOis propoWwwali* 
bus chordis^ hdu fidtim aUqudntulut^ in leomttricis txcrcimtU nom cfft po/- 
fi4nt. fUtc Mem fofttd^iudm upacris difcc {ft librt) prr d>ardam in uoct tonum 
tnfdi^rCyO' (^^ tomum:i^ qudmcwiqut uolcs in arte mufcaconfonamuim. 

CTntif dcm<mvrum uufccs finis. 

Arpionica rcgula : ioftrumentum cl>, quo cuji C rationis 
iadhibiro iudido)confonancix,confonanrian^quc par-^ 
't«,iD chorda pcrquiruntut,* Mclorum tna fuht genera- 
|Diatonicum, Chromaticum, Enarraonicura. Diaronicum 
l| geDUs:melos eft cuius parnciopcrfcmitonium minus, & 
duos toaoi continofc procedir. Chromaticum : quod per.^duo inxqua* 
Ii.i femitonia & ttihcmitoniumconfcendir. Enarmonicumvcro: quod 
per duas dicfcj confccndit ficditonum. Dicfishoc in loco,fcraito- 
nij minoris medietas cft:cx differentiae cxtrcmorum halwfudinis eius, 
parritione proucniens,qux 5c tctartemoria di<^a cft.fed vt carum fern 
per nuiof aux acutior,& minor quae grauior,rcpcriatur:neccflccft.Vo 
ces ,ncrui dK>rdc>rpacia:hoc ordinc a graui in acumen nitentia,in vnoi 
quoque mclorum gcnerc funt nuncupaca. 




Graeci 



nuncu 



CFroflambanomcnos 



P'".8P 



es 



Hypate hypaton 
Pa rhypate bypa torT 



Nuncupationes latin^. 
Acq uifitus 



Principalis principalium 



L icbanos hypaton" 



Subprinci palis principalium 
iTTdex principalium 



Principalis mediarum 
Subpnncipahs mediarum 



Hypate mcfon 



Parhypate mefon 



Lichanos mefon 



Index mediarum 
Media 



Mclc 

Trite ry neseugtTTcnon 

^ "^1^^ ^g^yng^gu gmcnon 



Tertio coniunctarum 



Penultima coniunctaruin 



Pjramefe diezeugmenon 
Trite Jieseugmenon 



Vltima coniunttarum 
Submedia difiun^larum 



Tertia difiundarum 



Paranerc 



nil 


■ Hi*!.- ., 

3^ 


Paranctc diczcugmcnon ^ 


Pcnultima difiunftarum 


Nctc diczcugracnon 


Vlrima diiiunctarum 


Trite hypcrbolcott 


Tercia excdlcQCium 


Paranetc hypcrbdlcon ^ 


Pcnultima cxccllcntium ^ 


Necchypcrbolcon 


Vltima exccllentium 



CMonochordum:cftquod vnica chorda continet modulationcm.Tc* 
trachordc^.quod chordis quatuorrPoIychordum vcro; quod pluribus 
chordis id continet quam quatuor, vt pentachordum:quodquinquc.hc 
xacfaQrdum;quod fcx.& itadc heptachordo,oftachordo,cnnachordo, 
dccachordo,cndccachordo,dodccachordo,tridccachordo,tc(raradcca 
chordo,& pcntadccachordocft intclligendunijquod vltimum omni* 
no quindccimconltat chordis. Proflambanomcnosrcft in vnoquoquc 
gcncrc, primo tctrachordoru grauiflimusncruusadiuftus^a proximo 
primitctrachordincruo,toniintcruallo diftans. Tccrachordum con- 
lundtumtcft cuius principium eft prarcedcntistctrachordi finis, Difia- 
(flum vcroxuius primordiahs neruus in vnoquoque melorum gcnerc, 
a proximo prasccdcntistctrachordi finaU ncruo,vno tonodiliugitur. 

CTctrachorda funt quinquc; Tctrachordum hypaton,Tctrachordu 
mcfon,Syne2Cugmenon,Dic2cugmcnon,hyperboIcon. 

tL Tctrachordum hypaton eft. CTctrachordu fynezcugmejio 



Hypatc hypaton 

h 



i^ahypatg 
Lyckanos hypaton 



m 



aton 



Melc 

Trite fynezcugmcnon 



Paranete iynczcugmcnon 



Hypatc melon 



Nctc iynczcugmcnon. 



CTctrachordum melon clt. 
Hyp atcmcion^ 



Parhypatc mcfon 



CTctrachordon diezcugmeno 
Paramcfc diczcugmcnon 



Lycbano s mcfon 



Trite diczcugmcnon 



Paranctc diczcu gmcnon 



Nctc diczcugmcnon 



N 



CTctrachordum hyperbolcon 



Nctc diczcugmcnon 



Trite hyperbolcon 



Ciij 



Para*; 



Paraactc hypcrboleon" 



i'araactc nypcrpoi 
Ncrc hypcrboleon 



CModum hicrocamustrcmiflionemaut intcnfionera oraniurti tetra« 
chordorum gradaticn in aliquo gcacrc mcIorurn,fui gCQcris progrcffio 
nemferuaas. 



C Modi Tan tfcptcm. 


A 




CPritnusHyfKHiorius 


-f 


t 


SiKTundus Hypophrypius 




X 


Tcrtiuf Hypolydof 




3 


Quartuf Dorius 




4 


OuintufPhrygius 




5 


ScxrusLydius 




6 


SepnmusMixoldius 




• 7 



CMonochordi regularis particioncmrin diatonico gcncrc dcmonftra^ 

C ^i'jwii Infiri^mmU muficcs qtumplurimd reperUntur ut Cyth4r^yTilU,Tu 
he,utui,Multtforjitilcslijlulx,Dextr^yLeut,Simpliccs, Duplices: inquibus u/ho 
fpinrw animxrUis C«r intuit Appalriwf ) fatur HUgnis Lydorum rex Uljrfixquc 
fdUr primus in cAncniomxnusiifcdpeiinAjt.pr^trruut ?elte,Chordaciftc,Sd^ 
lucifHyirduUySJ>ilue,?rdtcrii,f^igddeSyUsrbihySM4ld,v^^^ 
f4f(}fU.li^eUiTejliidoyB^b4ti,?Udr4, Monochorddj Tetrdchordd, VofychorddfO* 
titer d id genus muficd injlrtifnentdipldcuit amen Philofophis in MonochordiSy 
rctrdchorliSfdtque Vohcbordis monfltdndisxaterorum nos intettigere tdtionem 
dtque compofitioncm.quocircd utde mon^hordorum cr Tetrdchordorumcompo 
fitionc inteUigemusAtd dereliquis eft inteUigendumMunc ergo dd Monochordi re 
gulxrii nos cftenfionem conurrtdmus. 





■a J 


eff^hiklmnopqb 


1 


|t| 


S T|T |S T|T|T 1 S 1 T|T 1 S| T| T 1 


1 


^ a i 

r T^ 1 


c d 


e f ^ h i k I m n f q r 
T.Tonus S-sfmitonium minus. 


1 
1 



Mo- 



nil 3x 

CtAomchcrdum igitur rtpd^tn kdrcoJiicitttr.qucdin unko neruo mufce con' 
pnantU harmonica Yt^ulafuruiiti^fitur.sit ergo ^ b chorda: in qua uolmnus in 
diatonico gcncre conpmdiiias regulanin cpUocaTc.accifi^c flaniffmam regulam 
nulUcp tx forte fin^Umtem:MmUm]xnca ah.&Mplf^^ & dedmam- 
o^auam ficwndihuiusiab c inaintendo tonum. ^ op dine fimitoniunt minus, 
(^abein fjbi g inttndo dsios'tonos.(y ab g Inh fimitonium minus, ab b m i o* 
i in kyduos tonosjmrfus ab k in Itonum.^ ablinmfimitonium nnnus.abm in 
n^nin o.duos tonos.ab o in p/imitonium.abf inqO'^^T. duor tonos, ita 
quod condnle pmitonio minonfitbiun^fntduotcmtdemftisd quifrofrofiam 
banomeno primo femitcnio praepofttus efl tonuSi&l tonoqui t€trachordidie:(eu^ 
pncnijboc tji difiun^li ftiticifium eft, Kurjum afflko totam regulam harmoni' 
cdcr toti ehord^ a bita ut cftt cum a^O' ^ <^f*^ ^•Cr in chorda a b ubi affUcan 
tur d,«,f,g,cr rcliqui:e fiilionum reguU notai ftgnod,eyf,g^hyiykyl,m,nyOyf^q 
^ dko monochordum a b effe in generediatonicoregulariter diuifuvt, Qu^oniam 
emmfua ijiterualla interuallis reguloecrreffondent, euaduntque ^qualia:a b & 
dbfonat tonmn (^ db cr f b femitoniumy^ ittrum ebo* fb ionum continet, 
igitur ab^T fb duos tonos cr fimitonium minus continensifer fixtam tertij co 
fonat diatejJaron.Lt auiafb 0-g b ut in reguU continct tonum^^g bc^h bfi^ 
mitonium mimis^crh b CT «t,^ ib ix kb^duos Ponos.ergofer decimam tcrtij f 
bcT'kb tres tonos (emitonnwique minus continemiconcinit diapente, sed ^tno 
ftratum eft abcT jb modulari diateffaron^ ergo per uicefimam tertij a t.CT k b 
quod ex, conjonantiis diateffaron CT diapente confur^t coalitumyconfinat diapa* 
fin^Kurfits Kbcrl b(onattonumyl tftr mbfemitonium minusy mb^nbo* o^ 
duos tonos, quare kbcT ob tres tonos CT ftmitonium minus continens: ptr decf- 
mam tertij confonat diapente.Sed abcT^b monftrata eft diapafin con^nantia*^ 
ergo ab^ob confonat diapajon ac diapcnte.Vr ottered quia o t CT p fc femiteniii 
eftyO* p byqb^b duo toniurgo obo'b concinitdiatejfaron. k b igitur Crbex 
diapcnteydtateffaronque conftansiper uicefimam tertij con fonatdiapafin. At uero d 
bo'^b ftidem monjiratum eft concinere diapajon.ergo abCT^ concinunt bis di * 
apaidn.Uquia h^ec monochordipartitio per Jemitonium crduos tonos tal^aprO" 
celfity^" eft que modomonftrata fitnt confonantiicharmomcenguUjuffragio per-^ 
ueftigareiergo monochordi regularis partitlo in gcnere diatonko per diffinitione 
mon^rata ^.quodeft propofttum. 

C Mono^ordircgularis conftitutionem: in gcnere chromatico dc- 

i clararc, j 

Uiiij in 




- __g , ? 

K d e f e hi k I m n o 

T I J|A|TR|_S| A|TR|t|S| A|TR 




TR j 



k I m 



T.Towi- S.Semitonlum. 



op? I 

A..$cmumuum mms. TR,Tnbemifo«tM. | 



r^^Trr «t»?« ?ro pr<>fUmUnomcno irMungitur tonus :interfcritur que ubt- 
^^^mtHnmut fsrumcSnPTOionccntudifmnao tonus, hoc cjl inmono^ 
^^Z^n^'^i"f''>^i*'>'-0'^oaM4imnon4mcollocmr toms.Sttn 
ZTbu^vrms cbcriiw qu4 uolumus cofonrntUs in gtnrrc chromxticorc^Un- _ 

ZtS^mticunii buius c/towm:cr?<rdmmamoaMummpmdcrmt:^ 
2ri>wi.cr itaum p^r fnmm d f toniintnuJlum. mt ergo c frcmtom- 
^JZsfTdtcrci duoftrntonUftdcr 'iH^H'»-"o^ tomm^fimto. 
11^na^'<tus*<luamJttnhm,ton\o.a-$ hvhif^duofmitoiM:ms 
ZT(^ct2<\utmLus.a-i^t^t»*do4dtommvfi»^ommH mnus. simil. 
ZJ^V^aoK Ifi^tonus^cr I,>n,nduoremimni^,0-nomhmitonlum.ttre, 
Zum\L f>^ao diuifi'n 4PPlico ex ^quo line* 4 b.©- .« Iw«4 4 h [ipio conftmi 

.aodiJlineam 4 b # reguUritrr in melodu chromxticadtuifim. tiam c d to- 
nus erdftonus.vfgto'^'*' Crfemitomum minus, qu^re 4 CT S "'•^'»"" f^" 



Illt 53 



qaui% h I continent tvnum:^ I k trihcndtontum Igftur gddk duos tones &a^ 
fmonimnmbiuscontinensfcrftxtnmtertij moduldtur ikuffxron.fei cr ag pr* 
hdm eft didpente:ergo a k conftans exagC'^k diafente cr didtefftron per ui- 
cefima m tertij concmit didpafin^t fenitus eddem rdtione probdueris k o con/ond^ 
re diapente,^ k b diapdfin.qtMtre d o didpdfon dc didpente CTdb recrepabit bis 
dUpdfiniqudm conjonantidm nos trdnfcendere rythd^oricorum uetdt duthoritds. 
qui tnmenuoUtuUeriusconfcendereiex his que idm demonflrdm fitnt cr que poft 
ed demonfttdndd/ufcipenturj facile confcendet. it cum urn monjlrdtd partitio, 
per jtmitoniim C fifnitonium CT trihemitontum proctfferit y nift ubi integer 
ddieaus eft toms uttum didpafon , turn concentus difiwnSki (cruetur propria 
etdstpdtet ergo ex diffinitione fd6ium ejfe quod in chromdticp genae propontbd-^ 
tur faciendum, dtque propofitum^ 
CIdcmrin cnarmonico gcncrc regularircr oftendcrc. 



A Monochordum enarmomcum, 

] r I D I D I tt'I d I d I tt I t I d I b~| tt | p | p | tt 
^r^d ^ t g ^ i K I m n op q n 
V.Diefis tetdrtemorid. TT^bitonusT 



CSff ut prius chorda d b,cr c r reguU eidem ut i ceteris ^qud.ab pwflo c add ex 
teniotonuntyO* dhdadf fem'itonium mvmsi o'fpdcium d fpartior in aqualia 
pa notam e.eruntque d e fdu<e die/es.O' f$ fdcio duos tonos: qui ditonum im* 
plent-g h iutfrius duds dieftsAk ditonum.k I tonum. I m n duds diefts.n o drf o- 
num,o p q duas diejis^O' ^ r ditonum.qudtn regulam hocpd£lo pdrtitdntj<£qudli -^ 
terdpplicoline^dbiO'pfiiil^snotdSyaqudlidaue interuaUdin linaa d b deftgno 
per medids inter dCrb Utterds <l,f,f^, h, i, k, /, w, w, o, p, a, b. c;* quiddd 
tonus, ^ d e fdute diefts femitomum minus implenteSyO* fg ditonus: ergo a g 
tns toni CT ftmitonium minus, per decifHam tertij /onant didpete.ftd cr g b r dw* 
a dieps cr i ^ ditonus duos tonos cr ftmitonium minus continentes: per ftxtam 
tertij funt didtejfaronA^turCut prius )per uicefimdm tertijid k continet didpafon. 
Cr hoc pd£io monftretur k o continere didpente & k b didpafin z quare a 
confinare diapdjon ac didpente, c ddd b conjondre bis didpajon,^ quia hxc 
moduUtionis progreffio per duas dieftscr ditonum procediticonjtat per diff initio -^ 
ncm monochordum enarmonicum regulariterefjcpdrtitwmyineoque nuificds fitud- 
tds ejje conjoHantids cr propofttum. 

^ CTccrachordum hypaton.'in diatonico mclo diuidere. 

I Tr0f, 



M 



ProDambjno nignog A | A 

Hypare Hy pato n b_| T_ 

Parhypirc Hyparon c 1 S 



L ich i nos H y paeon d \ T D_ 

HVpac^ emcjoa 5 1 T 

Csuvaiofes-yionochorlidocucTunt ordinarc.lnfcrimres MdUmiretr^chordayTcn 

quibui iOTmtU'XMcrd ^uc awflW) dtfidtr^rcntur co^mfientur facH^f^cLonJlif 
tuto erzobcde tctrachordum hfpdtomim ut b fit hy^att hyp^ton, c fsrhyfoxc 
hyvs^n d Uchjinos hyp^iton.c HyfAtc mefon.cux ^auim^o ad iTMUXuiii ^^ntm 
4 ProL;itj»o/nfno«,cUda/rt i\^idcm qu^in tttrachordis non icmfutatur : fid 
vrimo adteaa tctrachordocuat yenUchordum,fAcio ergo intaudhmchordc a fif 
quioaauum ad cbordum t,cr [c[<iuutrcium ad chordAtn d,cr fefquaitcrum ad e. 
item fjito c ad d fe^quKh:\MiumJuo er^o tetrachorditm hypaton in generc dwro 
nicQiut dUUm efl)(lf€ dmifum.Sam per diffminofum a ad bfroflabammenosad 
hyPAt(n hyvatonicnt tonus. crq^^iaa^^d d lifquKertiumcftiUtdem fer dlffiniiionc 
pojambuHomenos jd iichanQn hypaton iorutmt diate([aren. fidv amiaadt 
vTo(iami>anomenos ad hypaunmfon fit fefqualterum: fcf idem d ad e confonat 
iiapentcpcr dcctmamtertum igiturtcrtti d ab e luhancs hyfaton ah hypatc wo 
fintoHO dtjhmiruf.Kttrfum quia c ad d fefquioaoMum cfiicrgocad dfinattonu. 
atucToauU a ad d dtatelfarifl^p-ab lofius eft,^ cd tonus 'Agitur fn6 rmi/6 
ad crdUuitur.fimitQnru minus, fjl imqihypatebyfatoad pjrbyp.byp./f/mtowiw 
minui^crVarkfVdtchyfatonadHchanon hypaton 1 nasiikhanos hypato ad hy pa 
ten mefon tonus. confbtuti^m ejl ergo t, c,d,e tetrachcrdum hypaton^per (hmtQni * ^ 
Mm mtnus CT duos fubiuna^i tonos procedens:perSffinitu>ft€min gen^ ^0^ 
uico paritno' pcnrachordum 4, b. c, d, r. sed incidit dubitatto for/an, cur 
nofha tempejlatemuftdduos tonos ad tctrachordorum partem priam^auim* 
que prtlocant: nos Mttcmfolum tonum qui efl prt^lariAanomeni utque hypa* 
tes hypdtoni Kefponfto peruia, in promptufjue eft : muficos noftra tempefta- 
te profiambauomtno alter am chordam tono itflantem pr^xiffe^ ^id primum 
memorant cresmumfdaitafje. 

Clncodcm dutonico melo: tctrachordum mcfon fubiungcrc, Scin ^ 
odochordo.a proHambanomcno m mcfcadiapafon conrineri. 

Profla- 



Proflamb anomcnoy A| A ""^ " ~~ 

Hypatehy p atog b | T B 

Parfcypatc HypacoQ c | S 



LlcEaoosHypatoH f | T 



Hy paccMciog cfT" 

ParhypatcMefon f| $ 

Lich anos mdog g) T 

Mcfc " ^hTT" 



D 



. H 

* *«f fffi^yhftctTachordum me/on,j-aao c fijquitertium ad b,crg fefquiodduum ad 
b,crffifqmoa4*(Hm adgimt oja pfr diffinitionem eadh hipate mefin ad mtfe 
diatefpnron.O' quidg rfd b tonus CT ftmiluer f ad g tonus. nam utrumque exjcf 
quUaaud fropmione ndfatitr.trgo pcr/extam terrijic ad f erit fimitofiium mU 
ms,eftit4quee,f,g,h,tetTdchordum mefin, per fimitonium minus CT duos tonos 
fr^edtnsiingencrcdiatonkodiui/um.^quUacpToflambanomenos cr hypatc 
mefin in praccdenti monftrata funt confinaredidpenteicr w {tefentUhbypate 
mefin cr mtfiyduaeffaron^ergo per uxceftmam tertij:pTofldmbanomenos ad mefin, 
confinat diapafin.contimigitur oaechordum d,b,c,d,e,f,g,h:confinanttam dia* 
pafyi.quod eft totum ptopofituni, 
^ CPrxpofitiooaachordo: in codcm gcnerc tctrachordumdifiunfta- 
r umfiibmitccrcAdodccachordum diapafogac diapcnre contincre> 
Proflagibaaomeaos A | A 

Hyparc hypacon b| b " ' 

Parhy pacc hypjtog c| c 

- Lichanoshypatog d| d 

hy parcmclog c| c ^ 

Parhypatc mcfog Fj ' T" 

Lichanos mcfpg g| ^ 

Mcie 



Paramcicdiczcug. kj Tnre coniunctarum 

Tptc^diczcugmcnog 1 1 Paranerc coniundtarum 
f*aranctc diczcugmcn. m \ Ncrc coniugaarum 



m 



Nerediezcognicgog g 

esit k / mnfetrachordum die-^ju^menon, quod CT dijiuniiarum dicitur, facto h 
ad k mtfim ad paioimefin ^ftune^arum fefquxo^aujm^ZT «<i wi paraneten diftu^ 
ilarwmfifqiutcrtiim.ad nuero netendifwn6larum fifqualtcram. deinde I ad m 
triten ad paranettnfaciofifquiodmdm.ftcergo mefi ad paraneten difiun^larum 

I ij con^ 



I 



(»»cmitii4te(f^on,o-di ncten dU^tc c(ier$o fn iednumttrtUmtertH: « 
4d n P^«»rt« ^ nctm.tom intnuMm.cr I -«1 •* tritcs ^c^cuimtnon,4d p^4 
netc^-.fvniUtcr tonus cji.pd CT cwn mfe ad f^anctcn "'V^^' •^'J^'g'"^ 
^dmfit mu,,crfimilit<r h ad k tonus -.ergo faftxtm tmnk ^ /,mt ftmtn. 
nium minus.mt itJ<,t^ f^ramft ad tritn, dic7cugmaonfimuomimmimsjm 
tc ad fsran^o, tonus.cr f^ranct, ad ncttn iirreu^ncn^, ."'^5'"'''^^/;?";^ 
quote trtrachordum difmniUrumfipmori o£bt c(>ordo m dtatmco ^^^"'"n! 
awH eft.it cum vxMtdtHi moufhaumt a h efft dUfafon.v F"/?"* '' n'W' ^' 
af<ntt:\ptur a n ftofUmbanomcnosa-netc dit7tugmn,0Hin J«'<Wj''^"^^^ 
ncoMtin%t dUf^H ac Sa^mtt-Quiafun nt,&<mca>iendx «fJ«. ««^!^'f "" 
4 chorda mtpccnumamH fanm,con[Htucrtqueuelimuy.fit tdtetracim^itmbK 
I m,quarunt meli,tritclynria,ffna,on,faTanctcpnricuimenon,nttclynt^uS 
mnon.fado^iJmlin ad n^cn fyncx,uimtnonff<imtcr^M.ciu0c Cr «'«"•"- 
tcm diatcffaron.o- 1 *d m parancten ^ne^eusmmon ad ncten, tono d^antem. 
paritn CT k rfd I tono.trit ergo perfexam tert^:h ad k mefes ad mtenfyneyug'. 
mtnonfemitonii minons mteruaUunt,^- k I ©■ ' « *«» «"«>Cr tttr^^dumly 
neieugmenon hoc cjl difwnilanm in grnere dUtonico ^ifum. 

CTctrachordum hyptrbolcon in eodem diatonico generc przdiais 7 
copuIare;& in pcncadecachordo,bii diapafonconfonantiam coplcn. 

3 
.1 
/ 



Arc 

bmi 



Vrofla'noanomenos Aj A 



Hy pate hypatov b 



cfaut Parhypatehypaton c 
dlolre Lydyinos hypaton d 



^ elami Hypate me/on 
f faut Varhy pate mr fin 



f\ 



G joirtut LychoJios mfjon g 
a UmiTC Mefe h 



d lantirc Mefe 



b mi PsTdftcfe jtr^gyg. k 



c folfaut Triftdie^cugmnto I 



cla mi N ftf 

ffdut Trite hyperholccn 



a Lpnire Sete hyfoholeon \ 



T" 



i 



b fa Trite fyn, k 
cfQlfaut Vdrdtjet c^n. I 
d idfolre Netefyn 



tn 



Tctra' 



rni 



35 



\retrdchordumhypaton\ " \Tetrdchcrdu die:(eu^.] 

I To, I Semi, j To. ] To. | Semi ) To. [To. | To. | Semi. | To. | semi | To | To. [ 



I Tf trudyr<fww mg/Sg I [ Tef r<<cbordti l?ypfrtotgo?i | 



fLsltn^f q tetrdchoYdt0H hyperbokcn.fdcio ut in frioribus n nettn dieTeugme-- 
non ad ^ neten hy]^erboUan,fefquitcrtiam:^ iccrrco ad earn cottcimntem diateffa- 
rofi.cr p <iiq facio tohi intcruaUtwt CT oadf itidem toni interuaUswt^fer fix- 
tarn tertij:n ado exit femitomum minus, eft ergo tetrachordum hyperboUon nop 
q^ex femitonio minore Jucbusfiibitt^^is wnisimgenerediatonicodimenfum.pd^ 
fer fr^cedtntemhmefe ad n neten dit':^€ugme%onconfinat diapenre: ergo hadq 
pi(/c ad neten hypcyboleonex confonantia diapcnte CT diatefTaron conftans ,con^ 
finat diApafon.ergo aad q froflabanomenos ad neten hyperboleoniconfinabit bis 
didpajon.^ cum totum polycbordum a q omt^noquihdecim perficiatur chordis: 
in pentadecachordo cr in generediatonico conftitutum eft bis diapapn.quod eft to 
turn propofitum. 

8 CPcnradccachordi indiaronica.mclodiac6ftituti:numero$ repcrire. 



l>roj'umbanomenos A | 



^ypatehypai 
Varhypatehyi 



ton b I 



^\ \i9^\ 



•hypanhyp4ton c 



^ychanoi hy^aton J~ 



|xi6| 






I45» 



H96T" 



Tonus 
\11664. r^cmLminus 



\i6l 



1 10368 

'\s74s 



I Tonus 



Uypate tnefin 



Varhypate mefdn f f 



I144I \iiSz\ 



I I I 



Tonus 



S(muminus 



t,yc hanos me[on 






I I .IP7^ I l777<^ I 



Tonus 



Taramefe die:(cug: K| 

rri udieXf^S^^^^^ > I 
Varanetedie^eug. m\ 



] |io8p (864 I \6^it I To;;^i 



1 1 
I I 



77^1 
l7^P I 



I 5144 



Tomis 



f^^e diei(eug. __«^l_ 
TrUe hyperboleon | 
Tdra netehyperboitof] 
-N^tThyperboieo)^ q \ 



I 81 lcmt 

I i- I ri4n4 



5831, I Semi.mmus 



1^184 I To;;ii^ 



Ir"' 



w 



T74l 



4«^l 

143^1 
rerti. I 



4^08 [rowTjr 

4374 I Semlmbius 
7888 j Tonus 

345;^ 
quar.toc^ 



Tonus 



I III 



NM- 



C smnai qm in nrnfica difcifUnd ft^cipue defidermtunfunt dafldreSi trifld^ 
ra,<iMdmfl4resji>cmiolij^fitrm,e^ <ji dHfU^pU^quadrufU, ftf<f$^ 
tcri ^'fq9itmij,cr fefqui^M^qu^rcd f\ dtfider^s coguofccrc di qucm nume^ 
Ti^m numerus mdwr eflduflus:if(km fdrthtfcrduoMqutm triflus.fartirefa 
truLdd qucm quddruplusifortire prr qiutuorM quern fefqualtcrtpdrme far trid 
Cr ternd au^c faduoM quififquitatms tpareircfer^ fy-quanioHgt^tru. 
ad qttcm fifquioOjuiiS'.parnre per nouem,^non4m aage per 8cr numeripa o* 
{btiwnfccunii anthmenccs ubique (urgent pctrtiifi maior daplus^riplus^quddrti 
plus,fclqudtcrJ^:fquiterHus ^t pfquioaM4US eft. it ft rurfum cognofcere dr/?de-^ 
fjisq^iwmvtus minor duplu hdieciplu M4ge per duj quetriplu: as^peryqui 
qiudruplu:a44^perqu4tUQr,quifefqudteru:partireper ijO^ mediemiddde. 
quern fefquitertium: partite per tridfCT f«ti4«i dJdc.qucm fefquh&Miumiparti* 
re per o^o^ o^Mum dddefUJldti'ft per tdndtm o^MdJim^o' dt^ninonest^c* 
gnofccs petitum.std nunc dd monfttandum propofxtum nos coftuettanmSfSit pen 
Tjdccdchordum in dutcnico genere m^o reptrtum, cUius nameri qt^runtund b 
cdcfghkimnop q.duc^ inftimncem duCftrid^qudtuory^ produCkam in mi- 
nimos tonifhoc eft uigintiqudtucr^n pCT^ minimos terminos toni: cr yemdnt 
infuuifioioco 4 t, qu^ per fepttTttdtn fecumdi drithmctices funt in proportiane 
fefqmo(\M4d,0' continentid tomm.Cdpio fefqwtertiMm nameri difitquc d^crf^f- 
qudkerumiquiftt e.crfi^uplismiquifit b. Kur^mftmofefquitertium numen h: 
qui fit m.crfif^^^^deerumquifft w-Cr ejus fubdupUm:qui fit q.quid ddddeftdi* 
dteffaron,^ ddde dupente^ergo per dccimamtertidm tertij ddde tonus eJU CT 
eddem quoque rjXxone madn tanus . ttft d o{lM4:ifn pjtrtem hdberet: ea eidem 
aSe{ldfdcerem c fcfqui(>{iduum dd J,cr c d continetux torum,modo dutem quid 
dcomperitur o^ldud parte carat: augco d b d e hmn q per o6io, /urgdnt* 
quetertio loco a b d e hm nq: qui nameri per cd?idcm feptimam admui* ^ 
cem ejndem fcrudbunt propcrtionem , quam CT nameri fectmdo loco poftti, 
quu rrgo d tertio I0C0 pofttashdbet o{iMum: Cd iptur adic^d ad djidt c, erit* 
quecddd (efqtdoildiiusy dtque cum eo tonam continent . itidem adieda oddua 
parte hddh jiatg-CT ^^duam parte mddmjidtlzT 0^ <^^d q dd qfiat f • eruut^ 
que idemtidemg ad hyl dd m, CT p dd q:fefquiodMii cr numeri tonorum.O'^'^^ 
h comperitur hdbere nonam pattern, ^lUm Migeo per odo q- ueniat k: entque h 
dik i^er o^.utdmlict4mdiaritbmetues fefqutod.u4us , tt fig o6iM4Jim partem ha* 
beret: fdcerem ffefquiodduum ddg.At utro qucniam e4 caret: augeo omnes uu* 
meros- tertio loco repertos per oHo ^ exurgunt in quarto Uco ah cdefghkl 
m no:pq:quiper eandem fiptimdm erum in afdem aduuiicemhabxtudinibus y ut 
Cr numeri tert^^ locudi^Sid igftur o£\aad eius parte ad g-facio fej' oftaua ^arte 

dd 



nir }tf 

dJf:f4ch eJkc ergo ftumerts quarto loco conJJHutos tffe mmnos fctdJecdchor 
di.fid4add efl didtcffaron, ^aadh totmr^^ cddd W9U4s:eYgo fcr jtxtam ter 
tijybAdcfamxnimn punas. ^ quid dhiflduf a fin ^dcdiafcnteiiygofcr ui- 
ctjimam tertijehtfi duuffkron^tp- $ hcrfg monJlYatifint [(fjuioilai^idtqut 
tonutrgoferfextam tertiju fl^ femitonitwt minus.cr codtm fa^locflendas k I 
tTfio tfftftmtonia mmord:cr c^etai adinuicem funt cognitl tonligltur ftntade 
cdchtnrdtdiatonkinuMcfifimt rtfirtu ijl irjim mtncrus prep^trbahcmini ad 
numtrum hypateshypdtotiyWHusicr hypata lypaton ddfdrhypaubyfMtctiyfenii 
tonium minus, fdihypata hypdton dd liduxnon iipdton,^Hchdniddhypdten mefo 
iuotom,hypdtes mefon dd parhypatenmefon'.femitonium minus, parhypdtes mefon 
ddUchdnonmcfinyCr Ikhdni ad mefcn,^ mefes ddpdrdmcfcndifwn^drum: trcs 
font pardmefcs ddtritemfemitonium minus, trius ad pdrdntten^CTpardnetes <rtfv 
ncteniduo toni.nctes ddhitcn l^pcrholcon-fcmitonium minus.trites ad pdranetm 
€r pdTdnetcs ad netcniduo tonueft igiturnotum propofttum. 
CTrcs duronicc diatcflaronconfonantixlpccics: aproflambanonie- 
p no ad parhyparen mefon concinunr.& quaruor diapenrc fpccics;apro 
flambanomcno ad triccndiczcugmcnon.feptcm verodiapafon fpeci* 
cs;iatcrpronambanomenon & paranctcn hypcrbolcon. 



species didtejjk. \ \ spedts didpenu 


•| \ species di 

Wpr.^fecTlf 


apafrn | 


rroflrmbd. A prt. | fee. \te.] \ pri, \fe. | ter, | qu. 


f-l4lJ.|/«-l/«M 
|o| o|olo|oj 


H>f 4.f3^. b Ta. 1 o 1 o 1 1 TO. 1 o 1 o 1 o 1 


|To| o 1 o 


Tdrhy.hyp.c Se. \se. \ o| ]Se. | o| o | o | \Sey\Se, \ o (oj o| ofolo] 


Llcba.hyp.d JTo.|ro. \to,\]To.\t0.\ o | o 


TO 1 ro. 1 ro, j o o 1 o 1 o 1 o I 


H>fM. me.e | | to. | fo. | (To. |ro, | o ( o | 


|To|to. Iro.jto o| o|o oj 


^djhy.me.fl ]Sc.\\ \Sc, Se,\ o ] 


|Sf.|Sf. \se.\$e se|o|o|o ( 


i^tchd,me.g \ i 1 II |ro. fo. ro. J |To|ro. |to.|colto!r,|r.|o 1 


uefe h] i 1 II 1 to, to. 


To|ro. !ro.|fo|fo|t.lr.|fo.| 


PdTdJie, k 1 1 1 II I fo.ro. 


1 to. 1 to. 


to\to',t.\t.\to.\ 


Trite die.l] 1 1 M | se. 


/-r.iMMrdA. 1 


Fdra.dirm] 1 1 M 1 


1 1 


\to t.\t.\to.\ 


Nttf dY.ii 1 1 1 1 1 1 1 ' 


1 1 r 1 |r. It.jfo. 1 


Trite hy.o 1 1 1 II | || 


III lAir^. 1 


Vard.hy.p ^ 1 1 | I | | 1 


1 1 II |to.| 


'^€teh^q\ M II 1 1 1 II 


1 1 1 II [ 



vnnid 



M 



C ?rimd f^ecies iuufdron: tono,femitQmo mlnou dtque tono conjlat.Secundd: 
femit^mo CT duUms Vomi. lertu: duobus tonis cr/>qi*e«t^/"fwromtf . Trirrtd 
Ipccks dUptntr. eft^u^e conft^t ex tono^famtonio minorc CT ducbustonu. Sa 
cuis:di^ohns tonis, fcmUcmo^atquc tono,TcTtid:femimniocr tribus tonis. Qu4r^ 
mitfibas tonis CT fcrmtonio, ]>rim4 [f celts dupafon : cjl qua conjhtt ^^jono, 
femi:x>niominorc,duohus<oiuf, fimis>nu) ??UnffrCydtque duobus tonis, secunddtfct 
rm:om,idobu$toniSyfcmtomo^tnbuit9nh. Tcrtu: duobus toms,femitonio, 
tT6ustonu,dtquc fimonio. Qu^^t4:tonoyfcrm:Dmo, tribus tonis, ft nuto^io.dt-^ 
qut tDno,Ciuintd:lifrUvnio,tTibus tonis, ferruvniOyduobus tom.StJ^itnbusto^ 
nuStmitomQAuobus tonis ,f€m4mio :: stptmu: duobus tonis,(eni4limo,duobuP 
toms}tnUtonio,dtquetono. irit itnqut ftr diffimtioncmi ^timxdidtejl^onfpt^ 
ati d ffrojldmbdnomcnoinlichjinonhypdton.Hdm profljpnbdnomenosdd bjrpxTf^ 
hyodton tonus cji, crhp^^^hfdtondd pdrhypdtenhyfdtonfir^tonium minus 
tjl'VAfhypau ucro hyfdton dd iichdnonhyfdton tonus, a-itidcm per difpnttio- 
mtniftcundd duteffdronfftcUs dbbypatc bypdton in hyfotenmefon rtp&ietur. 
Cr tcrdau pxrhypdtc hypdton in pdrhypdtcn mtfon. species uero dUpente: hoc 
paclo per diffinitiones fufttcntur . Vrimjiid profUmbxnomeno in hypaten mefon. 
srcundxidparhypdtt hypaten in lichdnon mejdn, Tertid'.dbhypdtemefon in p*<- 
rxfnefcn die^eugmenon . Uquxrt4:d pxrhypdte mcfon in triten die^eugmenon. 
It jeptcm jpccics dupafon : confimiiiterper diffinitienes qu^entur . Pfirn^: d 
prolUmbanomeno in mtfen. Secunddia pdrlrypdtt hypdton in pxrdmeftn ditTcu^- 
mcnon. TertU: dpsrhypatehypxton in triten die^eu^menon, q^4rf4: dUcbdHO 
hypaton in pjrdneten dir^eugtffenon, Qutntd: A hypdte mefon in mten dit^eu^" 
tnenon.stxtj:dpdThypdtc mcfon in triten hyperboleon, scptiniducroidlkhdnorjic 
[on in pdtdncten byperboleon.fcd ikcc co^nitufdcilufunt: mfpe^u ddi^entcr fu^^ 
^*rnore ligurd. 

CChromaticam princip^liurn urrachordum mcfonifubiungcrc i^ 



VtojUmi^xnomtnos A | 

Hypdte hy pdton^ ^1 Tonus 



Vdrhypdte hypdton c {semi,mi nus 

Xychanos hypjton d \ Apotome 



Uyffote mtfon e \Tnftmitomum 



^chromdtd dpud L^adcmonios induxit olim Timothxus Mileftu<,molioYemc4n 
tuntfuperiojcdutonico: in quo cdncndi moiohic tetracborditmbypatonqu^i 

mus 



nil ^j 

mus.sH er$o hcie tetrdchoritm ad hyfdtds in ehr^mdtkc mtlo amftauenids 
dfftffutum, mpoiw <imlm' 4^u^ fit ft^LauhdMmntmsi qumfdcmfkfyido^ 
SUudm dd b byp4tm hyfdttm fnmdm tctmcbordi c^flituem (horiam. idhit 
fddo d fefquattcfdm dd t iy^en mefontcritque ut in omnibus dddb tonus fCr d 
dd e didfa^eJdndcfddob d^fimitoi^um mhu4s,er bddd tomtm.cnt.^0€ 
dd d dficomccr qiM fer dccimdm tertij dentPto d b tono dbde canfon4nti4dU^ 
frnttirdmukwr di<^ffdTon.trio b t moiulMtur didtefjkron. Sed crnn ferfex- 
tarn tertij Judtcffaron ex duobus wms cr fimitxmio mm&re an^H ^Cr beared 
fimd fiMt tomsingo d e continet txmumcr fimitomm minus ^ igjitur d e trife^ 
mitxmmm . amergo b cbyfdtehypdton o-v^rhyfdte hpdton,fitfimttimum 
mnusyCT^if^trhyfdte kffdton&li^dnoshyfdtonfit/emitomum mdus^crd 
€ licbdnos hypdton tT hf^c ntefin tn^mitonmn ut mcnfirdtum ^:conftdteri 
go fer diffi^tionem tetrdchordum bcdeingenere cbromdtico tffc cottftitutmn. 
idem enim trijirrUtonmm ^ tribemitonium dicimus, 

J * C Cbromaticum tctrachordum tnclbnj/iibiungcrc* 

Vrofldmbdnomenos A j A {" " " ' ^ 

HjipdtetypJtott b I Tonus h 

^dri^pdtehyfdton c \ Sttm^minus c 
iJakmsbyfdion d j Afo tome * 



Mm 



tme fiu e ( Trifemitonium 



ff^emtfin f | Semi.mims 
Xkhanos mefon g \'Af>otome 



Mfff h I Trifirmfnium 



Csiwr pro tctrdchordo mejon in genere cbromdtico conftituendo efg h: fdcio d dd [i 

h dupldm cr concinentem didfdfon,& e ddfpmitoniwm minus, 0* idem edd g i 
fddo tonam, erit erro f ddg dfotomt.fed cum ddde monflrdtd fit conjdndntia 

dupente.cd igitur (ubjhdSid dbdh confonantid didfdjon'.fer uicejimdmfecunddm si 

tcrtij relinquituraidteffaron, efl ergo eddh hypdte mejon dd mefin : diiUeffaron. \ 

^ cum dutteffdron duos tonos cr fimitonium mifUds impledf, CT egftttonusr j, 

ergo g h continet tonum ^fimitonium minus, erit ergo g h : trifemitonium, erit | 

igitur efghex duobui fimitonijs t f Crfgf CT trifemitonio g h conjHtutum: in | 

genere cliromdtico tetrdchordum, quod erdt monjhdndum^ . 

C R c liqua duo tctrachordarin todcm gencrc prxdiftis adiiccrc & in j 
^^ pencadccachordoconfonantiambisdiapafon collocarc. 

K 



p 



Pronambt Domcnoi 



Par hypatc hypacon 
Licha aathypjrroa 

Hypatemefoa 

Parhyparc mcfon 
LichaDof rrtefoQ __ 






Tofittf b 
Semi, ml 



Apotomc d 

Tnfcmiton ium 

Scml.minuf 



g Aporome 

hjTrilcmironium 



g_ 



.Wamefediczcupmr. tciTonui. | Trite fyn.Seroijni.-^tc 
Tntc jier eugmciion 1 | Semi.mi | Paranete iyn.Apoco. l^ 
^ "^"^ I A^l^ [Netc fyD.TriTemiton. 



Paranetc dicxeugmen. m [ ;^P<> 5^ 
Neie diexeuy ncnoo n iTrif cm, 



m 



Trite fiy perbole o n o | Semuminut 
P arancte hyperbolcon p | Apotome 
Netcbypetbolcon g | Trilcmitonium 



o 



Csiwr r<» rdui^s mr4c\iefriis comflauiis k I m n o f q. quU tftrdchcr^ 
dum m9m iUmnSim eji d ttttdAcrdo netdrtnn dirnugmcnom karcofd* 
dcmefes aip^^fn dir^TMgwnwn SfiMum,4tt<mum. ^tmftndAn^^ 
ten iitrtufmen^n facio c^nfonarc dUpenUt cr 4d tieten hyferMiPn iafd* 
fen. crHetfdcborim kl mn fdrtm m teffddmdmH hyf4tPmtHr4thou 
di^m mftom f q f<trtiar at tn pr^eienH titrdchcrdum mpm. Lrttqueu- 
trumque J» Z^nm ckr^rndtico diuifrmM quid dddb cpgmtaeft efTe ^fdfm. 
Ubddq itida^,iidpd[^n.ergcd dd qc^findtbis didpdfin.qui^cumqmndc 
dm tio^s dtqiu cb^fdiscontiHtHmfttuofiftdt effcOum rjji id quod adt 
pr^poftttm. sedfi k I m fdds tetrddmdmn fynq^eugmm^'Ji^fumfdYtidiis 
ut^dcharium cbromdticum mepmCT fdcUefdaumintuebcnftofcftcym. 
CChromatici pentadccachordi ouracro* affignarc 



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C 5fnf t cr c mimmi rtumrri pmitQnij minoris, fer itdmamoOnuam fccufnii 
htam Tcfcrti. quia b nona farte caru: Migto b ^cfer noue ^ utmant b^^ € 
injicundo loiO,quonidm igitur mfccundo i^co b nonam forttm habet: earn dug€0 
fcr OS0 cr fi^ d,tunc b ad d:pcr jeftimam fkcundi arithmcticts tmlefyuiQ^ht^ 
uus. quote b d coatincmu tofwm, o^b^cfer candtm eflfcmifoHWM minus* 
€7^0 cd(fl dfotomc* Aurfrm quia b babit o6biuam,adiiciQ ttdtmf^am c6biuam 
^ fiat a : rmquc a ad bjefqui^^uus atquttonuiy^ quifmam a habet tcrtuunz 

faoc dad c f<pjuaUaiim*c/'^^^*^^ f'^^'i<i^i^'4**^ 
fericitem b b^iere noiumtpUttiam CT jccundam. facto tfftwbfuquioiiauum ad 
k:Jcfqualtcrum ad n^CT dyflu ad q.quofado duco bvccundiloci inabc dch)c. 
nq^ uemaat m tnho ioco abcdebku q,Ddndc duco c dftcundt iod in efe* 
cmdi er ueniantfifO- inker uemant I m^cr inno" nmant of in tartio loco* 
€ri$mqnc fetfcftimam CT odnuam jecim^ arttbmetica numai fenadcaubor* 
di cbrimatki m tcrtio loco afsignatlMam abet^t tonus f^bc fcmitonium mi^ 
nus ^cd afcwne^^ quia 4 ode eftfcfquaketfa ^ e (unt nu^mri diafente^ 
a b numm torn Jem f to igitta a b tono^tiinqmtur b e diatcffaron. ^bc^c d/t « 
mui fitni tonus A^itux ftrfcxtnm tertij defimt trifhmtomum.tt ah eft diafafin 
ij at Safenie^ iptur t h ejl dtattffaron* CT ftt o£biuamfecundi arithmtttcts e 
fCT/xy*^'i^wttai»wmfmw<j CT^f^tomeApturghcfttrifemitowu.cr conjimi^ 
liter fkonlhMs bk effe tomwt,K l^lm duojemitonia^ct m n trifemltoniumfn 
ofduofemtonidyetf q trcfetnitonium^cUfUm iptur euadit popofttum. 

j^ CModorum diatooici & chromam parhypacc parhypatis: pararacfe 
paramclts^acquc rrice mti$ corrdpoBdcDC. 

C Nrfm In utri/que ^cneribus by fate ad fitAyf ams femtonii minoris obferuant- 

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intenuOgm.O' mft^^fsmmtf^ tomm. Cr ^tfj^ tr\tai minus ^mtohm 
tfkmtiir neam iUs uocnUs in Btn<^ cioiemdi m»d9JM tmxct rtfpowfcrt.er*** 

vm utnbi<picmfimb4Momm4d byp-oj hyfooncpncmtnt tcmm-cr^i^ 
pjmimtfiniupmtt^dmpis dijf^. 4d ntms Hfxmibtmm &tp4pnac4u* 
foae. a- 'id *tm hyfetbokon fcu diafafon. tji tr$9 qwd frofontbAtur ©• am 
fliui: facile cogmtttm. i j, 

C Pentadecachordum cnarmonicum conftituere. 

Troflata banomcnos A I 

tiypatt hypaton 



h I TomlJ b 



pjrhypate hyfaton c | 4if/» c 






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1 .tTaiicU bypfftwlton p I dtf^ 1 

jjftfbyptrb otfo* q |7ifo»a^ ^ _a_ 

CHi.wi «>iiti«» Pfr/^inlu (/J./orio «iJ»:»t w c^f"^ prrf«do.ttki,4 4d {, >«- 
t,r«4lI«M »o«i Cr <« -d f duPtnU. Crhadd facio fimtonium miriUi. V dtuiio 
chotdac in medio dtffmntu b od d-.crit (tgo b ad c ditfts m^m»ma axqut qua, 
^ripjTtialU.parUtr ^cadd ditfts. ftd per quartern ftcumdi hmui h^c Utm^lU 
uerocontuaitT. ntque per triceftmamficumdam eiufdem: femtomummms m 
duo <tqua certo,co7iftiiutoqi tumero diuidi pottji.tr a ad e dupctt.fubfha£io i^- 
tur d b to*to:nlinafurp<rcorrtMu dtdmt tertif huiut b ad t tffe duteffaro-o- (u 
iuttfTarortmitomi minus ttduoitonos copleilatur.cr h <*d djttfimitomU mmus 
uMnquitur i^iw dadeeffe ditcnus. eritiptmbcde hypates hypaton,pjrhy. 
pates. hypatoft, [idMnJ^iff hypaton, & hypates mefon, tetracbordum per b c, cd, 
trdc dieftm cr dieftm a-ditomm procedens : per diffimtienem m enarmomco 
genere Jkuifum . sirmtittr tonjiitua* efg h tetrachordum mtfinfacirndo a adh 
diapdfin. creadgftnUtonwni minus. CT partitndo mediam SfferentUm per f 
i« ut t fv fg f'ft due diefes. Kam abah confonantia ^pajonfubdua^ a e to, 

jonantia 



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Jondntid £df€Hte:rtlmquHur$^ht{f€ ii^itefftron. fir cum egjtt/emitonmm nU^ 
msr'tgitwrihait HtmMS.tr cum t^fcrfgfimdu^Jie/isO'g bdtowa^ergo 
hactttrddSarH fiOrtHhringniertenarmdnUofdAic^fffl^ur* £t faibtbmefi 
ad nttenJbfiMm&miiiaf<nttitr *ii Mm Ipocrbclem mdfa/ott,cr c^rda mtfi a 
pdrsmtefitctTd(bmdtdifimn&ai%mtimadifcbipt , femdc ac frofldmbdnnnefto 
Jf hyfatthyf^ftom: fortirris tetroAwdum dijimidxtrum klmnutbcde tetra 
choTdumhyPatan.CTt^tT^^^<*ydum no^q hyfcrbcleon exctlUntiumqueiut e fg 
b tetrdchordum mejon.tctrdcbotdum dutem coniundarum b k1 m: fdttieris ut re 
trachordum mefin^cr ^ quoque fdcie efl. 

^ CPcntadccachordi cnarmonicinumerosrcolligcrc. 



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C s int fHttnnl dbcdefghklmnopqin primo loco cotiftituti nutncri pcntadc^ 
cddbordi dxdXonki pa o^taudm huius reptrti. pono itirum dbehknq inficundo 
loco cr inrcfpondenttbus locis rurfumcin fiiwndo loco ttdtifmutodd d, CT fdd 
gy^ I ddmcro ddp^quibusitd in ficmtdo loco difpofitisitx ipfts qutdem infi- 
cumdo loco capio differentiam b dcT ««^ mcdUtdtem ddncio dd d cr/irff c/iwi- 
littr djffcrcktum eg CT medictatem dddo ddg Cr ^^rf-CT differentidm km a* 
medictdte differentia ddk£id dd m: fiat I-Cr diffcrcmiairt « p CT medietdteddia 
{id ad f:fiat o.dico dbcd ef^hklmno pq numeros pcundo loco pofitosieffe nu 
mnosvattadecdcbordi endrmonidi^d d tmrrow<J.6r,crc<Jd«*« diejes conjii- 

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€9 nSkuinta fmitxmbm mims I i^B^ «*» tflhcin £^tuncc:f4&tm^ I i 
mcnjrm9nU0.crde€ftdit0mis.UMVtodifiindUtomc0Cc:dthub^ 
mc9i€MJUdt9mu moiuUtmuctikonusefi.ndmcrniamucdtofmmo'd^ 
Hm n. cr hpc f^eto ufmaKtifr t f j iM hdis CT^ b iix^mu b kr©iM«. k I m 
dutibffis. m n ditonus.nQfiu^ dufis.f q iitonas.suMt iaquecJUatmimeH 
ftMtJiec^iamisaidmmiild:qw^ pra|H>/irw. 

CModi diatonici parbJiypjacfurmooica modulatione trifit inlicha 
Qoniflcparh^pare mcfonin IichaQoaracfon,arquc trite ioparaacras^ . ^7 

C I J ^ffidjiatim jr^mftum ^.sam in di4tomc9 ^ctddtcdcbor^. d fr^fUm* 
bsnomoio di hyfoten byfdtpn, tnddit fmi. CT ^ ny^^t dd foriyfdtcm *yp4- 
ton: ftmtfmiimmmis. in aurmomco uero d fTcfUmtfdfiomcn^ ad byfdtcm tio^* 
fjtcufimi^itff imdibg tMUi.cr^ h?^ ^ lidkmon hyfdton: ^mttQwrn mi^ 
ntLi,dut9med rptur fdrhyfdtciin enarmcmcd moduidaam fermumnn, ttdnfttcf 
in IkiKOiwi. tT ^^ f^^ dc rttiqids oftcnddtm. quod CT fdctUfcnjii deftthcnde* 
fur: ft m utf9qu< emodttUndi genrrt CT didtomco cr <narm * mco,pr9fidbdmmaU 
fondmiire<iutfiid.tHnc cmm tndMiftftum aicbuius lidxwof uit»i fdrhyfdtUtjfc 
^uxfinoi^fJ^nS^qite.fjrua cr ^J^w^^ fdrdnctdixJiiui rritii cf[t ^uifrnas. 

CCoafcaduat in tribuf modulatioautn gcncnbustproflambanomenl, 
hypatcpriQapalcs,bypatc racdic,niclc,paraaicfc,nctc,cum difiisdcy ^8 
turn coatuaCte>atque cxccileanbus octis excellences. 

€[Sdm in tribus gcnerius fa^^^udndm,quinmmy/exmm,fiptimdmfdccima, un* 

drdiiMffi, iui>dccSpM« cr ^f^AM^tC^^ 

hypdtonkium tonum.o' I*'*''* ^d hiyfdus mcfit:fiHdHt iufke.ddmepa: dw* 

f^tfinMnms difiimilMizfiiuMt dufdfi>t dc didftnte.dd nctds dutem conmndas: 

4cmM twtofiiunmdidftfin dC didtcfpmm^cr ^d ncms exccUatttii confiuant 

tAiiiket frofldmkdnomem Us Sdpdfon. cMJintmnt igitur in triims ^encrHms qM 

addt^ac font t^octde.qupd eft ft^fofitum. 

€EQug chordg mobiiet q^ vcimobilet ia pccadcca.cxiftit iue(tigare> ij 

Proliamt>aoomcni A | Sttbilci 

Hypatehypatoo b| Stabiles 

Parhfpate byp aton c \ loftabiie s 



Parfafpate pyp aton c | imtaPiie s 

LicEaai hypat oo" d \ laftabilct 

HZPi ' . - - - 

Parh^ 



ate tncfon e \ Stabilei 

loCbbiief 



Pat by pate melon 

Lichaaimefon 

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f 



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Stabtlet 

Para- 



^^ ^ ^ ^^^ 

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Trice dieicugmtoonl j InfabileyPyanet^ frtiezeugnieJnftaEilcr 



Paranetc fljeicugmcm 
Netc dicrcogmenon n 



Inftabilcs Netc fynczcug mcnoaStabilcf 
Stabiies 



Innabiles 



iTritc hypcrboicon o 

Paranetc hypcrbolc6 p | iHftabilcT 

Netc hypcrbolcou q | Stabiles " 

e.ChcTd4sftabiUi immobdefque uocamusiqudc in omm fcntadtcacboxiorum diui 
poncuicm firuam intmuuU^andmque ad frofUmbanommcn babitudmenkin 
ftabiles uero atquemohilcs:au^ idnon/hudntfic trgoflahiks acinJlMcs refc* 
rUmusxfmadmper'fr^ctJiaitcmmtmus gcneribus confentidnt frvflwib^tno- 
maii,frbKif4dcs hyfate^medie hyfdte,m€fe,fardmefi,netc tumf^nr^ugmau tu 
diticugmcne 4c }^fnbolcs,0' cadcmuttxdmonjbratis iamfatetfiruent inter 
uaUaiermitergo frcflambMnomfnoSyhyfdte l^fatcn hyfdte mtfo n/ne/e^faramt* 
fi,ncteJynq^upn€non,nete di€:(eugmenon.CTn€tchypcrbole0nfimflicitcr immo- 
biUf dtqueprme. Scd cum 17 huius mot^duerit ^arhyfotds didtonicc moduU* 
tioHiSfin enarmonico mdo ttafireffrmutdtique in bcbanaSjO'^i^^^ ^m fdranctasi ■ 
confidt (rgorcBqudf d fticdiais tffe mobiles, utfarhyfatcn hy fdton, Uchdnon 
b)'p4f «»,prfrlyyp4ro» mcfonylicbanon mefrnjriten fjynt^eugmenon varanetcnfy 
nq^fugmfnon triten ditq^fugmtnon faraneten Ar^cupnenon tritat bypntolcon, 
fdT4nctcn hyfcrbolco.id tdmt dnimddunti dignum eji quod cum per 14 huius far 
hyfdte c trite in didOmkis crchrcmdticts reffcndedntfinnlidqU4iepcfpdednt tn 
tcrudUieds non ufqu^ ddeo(ut c^ere funt)ej]e injiablcs.^'poinae fartim mo* 
biles, fdrttmqu^eimmcbiles'bQnoiwrt did fojj^e uidentur. 
jQ CIn tctrachorcTis diatonicisrabhypatc hypaton prime limitcconfo- 
nantiarumad quartum vfquelimiremter diateflGiron continetur^ fed 
femel duntaxat in ftabilibus imniobilibufque. 6c ad quincutn limircm 
idcmtidcm rcrdiapcntcfemelinmobilibuS) femelpartim TariabiliSj 
& femel penitus inuariabtlis atque firma.ad oftauum vcrofepties co- 

fonabit.dia pafon:terin immobiiibus,8c quatcr in mobihbm. 

Hyodtehyf^lQ- confequentes fuis locis. 

I sent. I Ton. \ ton, \ Sem* I ten, \ ton. | ton. \ Sem. | to \ to. | se. | r. | f . f 



ufOOes^ c d f g, I m op 

C Cum ikitur a ^^^^ l^^^^ di quartum ujquelhmteniyquintum, dco&duumi 
frtnws Umes nonexclMditur,ftd quartum, quintum,ac o^aaum excludi intelliff 
mus.sit ergobcdefufque adq quatuor tetradiordaiO' h hyfdtehyfaton,!C4gtm 
ucro conpquenhs. Vicofrimo d b ufquee ter contintri ^ateffaron: fed /tmel 

kiiij lolu 



I- 
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sacumJm\^umtLm<iu^cpmicf,tabbyf4nhr^»4dkn't*nm. 

ttrdJ4'ironMfifntlltntM4tmjiJ^usjmmAmu^<{HecontmrtJtcu^ 
dko4i>hhy94tthyP4tondifi>f<{iu faA,yfa:tnmffomfilim ttrcontmmdUpt» 
UM^m pcrdtdmam urtutiUfa^t ttibus tonis ffmit<moque wwore coji^.fid 
per quorum erii'i'^'"' bmu'Jfddffilim duos tatws c i-o fimaoiM mmora 
conMct,quaduofmUoHUmomdfrr}^pcundi:mmiuwm>t<,m>,comm^re* 

flitUKnt.trto ut b adfdi4feHtt conjonsntum comfltJt.flau dttfl afotom. wn 
timfiUuitmo b ad f diofottt.um cadtftr quort^m & quiMtfmhuws comi- 
ncttrts tones qu\ funtcAi d,dad c,crf^S,<T t^nufanUonmmnutqHod ejl 
€dd f.irititTperd(amdtrrtiit4dfcofotuitdupete.erit<pc4difnmd dufttttr 
per LQc d h h cSciiut 3 an.quifut d4dt,fadi,cri4dh crfimmmmms 
t idf.ergod 4dhcSfin4t^pete:eritqutd4dhficudd dUpat.tr « < ** Km 
A, < Cr 6:anifttmixtet contmentur 3 ton j cr fimtamm minus. erUet^eddk 
tatu dijpeHti.ter tptur db h ufqut ad f: coHtinttur ^ptnte. fid cum priiiuftt 
^ c parhypote hypsm we Iicl>4i»<»ii mefiM,cr p^Jhypatus cr Uchdnds 5»«!F'*.' 
uerit dteimMOHd huius tffe mMts-.mterstc ddgprnru dupaUeinmomfus^ 
conllUum.cr amficunddfit abdUchdnofHndpdiiim in h mtfin,M>MosMtc 
moLmsfit/ymtfimmokitLs-.licmtddisititr^pattepdrtim udridbmsexiltit. Et 
cum tertid fit^itc hypdtt mrjon dd k pdramejin, que per tdndoH itcMuninona 
ftjtO*/ ttun^dttfbnt.rriisiur tcrt'tddidptnttommnoflMisatautfirrnd.rrf 
n^cPtirb \typdtt \rypm»i^ut dd k pdrdmelinfcptUs contineri ^pdfin-^id /3 
lum ttr in rmmobilibus/judteTdutan in mMibus.Hdm bddk ptrquartam/juf 
mw crft^t^ •>•««« coiabntquiHijue tonos cr duo frniamd minord.trto per ui 
cefimdmprtmdmtertii-.bddkcoM&njt di4pMfin,entqutb dd k printd ^pjfin. 
E t per ide-M cddl^dd m,o- ' ddniftnguU interctpiunt quinque amos CT daofe 
miamid minord.mt trgo c dd ICtcmtdd didpdfin. criddm tertixcrtdd n qusr 
a.fed per qudrtam,qimttm, [exam dtqut (eptmat fmdifdi ofimlittr conti- 
ntM quinque cottot & duo ftmanid minordfimiliter tfgdipchdd q.mt er 






"" 4T 



^ qmmdidfdfin fad c^cxmg ddp.crfefthm h ad qj^tur dhhddk ufquc 
contiHCturftftia didfafin^atqui frimamftabcnt hypate hyfatvn cr foramefc 
dir^eugmenon.cr quammhyfote mcfin & nete dir^tugmmon. trfeftiTttdm 
mcfc cr nttc hyperboUonJbypOas autm mefis^fdrdmepts cr netas monjirduit de 
cpni4oa4UdimmobiUsdtquc(mbilesJgitur inter ilUsfeftmdUfdfon confindnti* 
^tdidpdfin ter in immobilibus reperitur. Scdfccmdd prdchent parhypdU hyfdton 
Cr trite die^tupnenonittnldmhchdnos hypdton cr pardnete Sq;cugmeno.quU 
tdtn pdrbypatemcjon cr trite by perboleon.fexmmlkhdnos mefon cr pdrdnett 
hyperboleonStddcdmdnond pdrhypdtds^tritdSylichdndSjCr pdranetdsimonjird" 
uit ejji mobiles Agimr inter iUdsfeptem confondntia didpafon uicesyqudter in mo^ 
biUbus mutd!Hlibupiuefa£iareperitur. quod efl totum propofitum. Quid duttm 
diuus seuetitiusfdpientum IdtinorumdifcipUndsfeadntium primus, quern CT in 
hoc opere quantum ualemusimitamurfiiis difcipiinis nonpdrumddiuti, decimeter 
tiocdpitequartifui^nMfices, [pedes didpente numerdndo eds qudtuor fdciat, id 
introduaorut fd6lmn putetur,ubipr<edfdmnoncurduitueritdtcm,fedcommt0tc 
fecutusiUicejlcftimdtionem.quodfdcile ex eius fuperiori determndtione cognof* 
dtur.hic dutem non vntrodu^onis fed exalte determindtionis locus eft. 
CTcflaradccachordi chromatid inter afllgnaros limitcs tcr itidcm 
^ diapcflaron folum fcmcl immobrlitcr.bis diapcntcrfcmcl immobiliter, 
(cmclcp partim mobilitcr.fcptics autcm diapafon: ter vtin diatonico 

generc immobilitcryquaterc^ mobilitcr continctur, 

I ^^' I ^pO' I Tn/r I se. I Apo. | rrife. | To, | se. \Apo.\ Trife. \ s e, | Apo. | Trife, | 
1^ I I ^ I lb ic I I n j 1 -^ 

I ^ ^ f g I m p I 

e.Te(ptradeaichordi0n ex qudtuor tetrdchordis conjhtmturiicarcoftc nuncupatu^ 
2> qudtuordecim thordiSyneruiSfUOculifuecontinedtur.dico enim primo quatuorde^- 
dm chorddrum ingenere cbromu^co continue per literds t)C,d,e/,g,b,k,I,w,w,(7, 
p,qddhypdtehypdtondifpoftmru: d primo ddqudrtum ufq; limitemjter cotineri 
didteffdron (olumfimel immobiliter. i^dm per decimam bdde concinit didteffdron. 
eritq; bdde primd didteffdron. fid o* ^«w per edndecdddeft dpotome CT dddc 
trifim4tonium:er^o cdde continet duos tonos.O' f ^ wndecimdm e ddfeft fimi* 
tonium.ergo per (ixmrntcrtij c ddf duos tonos ^fimitoniumccntinensierit did- 
teffdron. erit i^itUY c dd ffecudd didteffdron.p- quia rurfus per wtdecimamedd 
f eft fimwmi0n minus, o- fad g apotomeiergo e ddg toms .fed ddde monflra 
tu^m eft trij€>mwnium.ergo dddg continetdtios tonos crfirmtonium: eftq; dadg 
tertiddidteffdrori. it cum primd b ad efit ab hypate hypdton in hypaten we/Sn. 
per decrmdmo6btUdmjit immobiliter.ficfvidd uero c f,^ tertid d gt/unt a parhy^ 
p4tf CT li^^^^o prindpdlitiyqu^ decimdnond patefait cffe mobiles. coftdttrgo pri? 

L mim^ 






M 



mum-sccmio iu.jUm his uOr, <i,^tumUmtm ccnUmrt ft^'-V"."^ 

m» pa dccimxmtntidtertii: 1 4Jg<»»ci«rdMportf. trit<iu<b^ jfrwi4 4^ 
Jutlca «iw7«.«t« cr MS qui$^rxflthim:j^prinami>MPmtn^fn- 
uans-MCddtnoM expUtAt duftntt.tHimftrm«d»m<mfiratmui4Mfaacto* 
flmouum dtaU b A cfimitmum mirmju^pcad h. Nd« IT ""^^tS! 
tadb cnrturt trifmitMium. atc^ad dUfente comflmentu plum ^fi"^ 
tomi mvuiifuftratrntmch ccnfon^ntU diaftntt t^fUto tono^ntqid WwUi. 
Mt dupaucsam tic h fuptrtt conjouMtiam dUf<ntcinttsrc,tomMo<pton^ 
dim att c dapctomc rtlvtqMurdhJ'uftTMS disfttc confonMtu^mttomo mt» 
imtSeit4d k «w/5>«tdupfrc */i own b *J k f o- dHodtdmatonui.ftdd adb 
per imHtduUcmonjhati (iprr^t confonMtiam iutfcte ftmitenio mmore. demfto 
m» d t ttifanitnw trifanitomt minorr.rtlwqKitur e b dejicies f<?«o- dupm^U* 
duo mtJfb k uno firt duifHt.cfi it4<p t ^ k-.fccumd^ dwprt*. aJ. b iffft/^^l 
qui»^ limitcm-.folum bis fumitut dia^cnte. tt cu frima diaftntt b$ftt<ib ityfm 
hy paton immobiU, dd lid^non mtjon fartim mMm: fit ergo pimo fartimm* 
biliter. Atuao cufHfccuHddckfitdbhyr''te »tefonadf4ramefen,qu<emonJtrate 
Qnt immobiUi.fit mo fiatmdo mode itnmoUlittrftruaturqiftcundo modo dutfcte 
vrovriemi ut dt QUtnto loco in aiiytti fut locum.Tertio dicofeftiti fieri dufd^ni 
tfr lwM#WiftT,fl«4frr uero mMUer.Hdm cum badefcr frjmam fartem hum 
monjh^tafit di^tffaron,v ftrftcunddm tsd k diapntt: trjo pfr«kf/iw<tff. 
tiibdd\i confitut dUf^dfoH.erit icaqutbadk frimadu^fin.vr4ttert4qiMC^ 
ffuit iuuentnfccunddduiteffaron, V H4 d^otome^b tn^tmxormm-. ergofb 
duo funt tew cr b It toms (y k lCenutomum.eri»f\dtdtcnte.cr cfutidtn <Jiftu 
W? didttfftrS.coHcimt igituT c dd I didpafoH: eritquc c I fecunda dutfafin.& eodp 
iure d in tertid duxfufin ddiaudnte duodecimd huius.tr t«<l>*^t^f'>'i'**f* 
g pffxra. crbq feftimd. sed ter fieri immrrtobilitcr cr qudtermobiiiter:cedetH 
modoutintOceftrndmonPrdtur. , 

CIn tetradccachordo enarmonico:interafsignatotlimit«tcrdiatcl- »* 
faron & fcpties diapafon,vtin'prxcedenribusconcinctur.atfcraeldii« 

t a<at diapentc atque immobilirer. ^__ _^___ 

\ Dif. I Dif ■ I Difo. I DK- I Dtf • I D'fO' I "ro"«^ I D"- 1 P'f- 1 P'fo- 1 ^^- 1 ^*-\ ^'^^ 

|t> I I t I I *> K I I » I I «f 

led ^ ^ , . . ' "* ", / . 

tijetTidcacbordum cr t^jarddtcaihoTdum-.idem iicimus.jitiptur b,c,d,e,f^, 

\,,V. ),,m.,n,e,v,q,ttffd:tdiecdchoirdum tiutrmonicum. dico ^mo interb e tcrccn- 

tineri didtejfxron.vnm <juid pa dedmamquintam hutus b c dfuntdu^e diefes te- 

tdrtemorie: rit b dfemitomu mimts. zxi^'j^ ditorMS.erlt i^itur b e priau dia- 

tefftto 



! 

Ulpron :tr toim iun e ffxmtid.& dg tertU. trquUhe finthy^dte hyfdUm 
tf byf^itc mefin: fit tgtturfrlmdimmbiliter, & c ifimt ^arhy^ate hyfaton & 
licbd990S byfdtm mMes:fitiptiir Cr duobus modis m0biliter.secud0 dicofclufi* 
wrfd pimo 4d qmtmn lifTdiem fieri dUpentr.& id quidcm immoMiter.ni nam 
pet Mdjfente bfiqmd fildrnfuferadditconjonaittia didteffsron b e diefim tetdtte^ 
morumMque bgiqwdpAumfu^eraidit coy^wantle didteffartm duds ditfii ,qu^ 
fitm femitomum minus.nequefiet b biquid conjonanti^ didiefptron fu^erddditp « ^ 
ffttttnimH minus cr duos tonos.neque fer idemfiet didfente eg dut c kndm hie 
dbwHddba toruisc diefistiUkdUtemdeerit t^ms minus und dkfuneque d g.ndm 
detrUtoms,neque d h:ndm tonus AundSit^at uero cum chfit didtefp&on. ndtn 
t fg due die/is,crg b ditonus, O'cymhkfit tonusierit igitur e k didfente. cr 
cum efithypdtemejdn cr k pdrdmefi,quamonftrdte fitnt immobHes,fit igitur w» 
fer dfiigndtos limites: folumfemel didpente dtque immobiliter. Tertio ft f ties fieri 
didfdfom ut in didtonids: ex dedmaquinm decldratur, quemadmodum ukeftmd 
huiusmonjhdtd efl. 
^j CDiatonioc raoduljtionisrfcptcmmodos ordine collocarc. 
Abcdefghklmnop 


R Hypodorius | | | | | 1 1 I I M 1 M 


-J did 


S nypophrygws \Tom4S { | | | | | | 1 | 


q to 


THypolydius |se;^fa.A | I | | | | | | | 


1 1 


1 <ini 


V Doritis JTonus A | ] j ) | | | 




]\ \ \ \ \^ci 


XVhrygius \Toms A | | | | | | 
y Lydius \ scnutcnium A | j | | 


- 


1 11 1 !«• 


1 nrirr^i 


Z Myxolydiui | Tonus a | | | | 




1 1 1 1 1 II 


A b c d e f gh K i m H f q 

€.Sitdb cd ufqueddqpentndccjchordum didtonictLfit r pro hypodorio pentnde* 
cdchordum diatonicum abcde fghklmnop qgrduifiimum: quod ut aliorum 
bdfis dtque fund^mentumfidtudtur, cxtendo unotoftoin dcume profldmbdnomc* 
non peutddecdchordi s dmplius qucimfit proflabanomenos riad quern c<£terds uo 
ces fuo ordme par quartam ^fcxtam huius^m didtomco moduldndigenere/ubiun^ 
go.eritquepentddecdchordum s per diffinitionemihypophryg^ij modi, ftmdtct ex* 
tendo profidmbdnomfnon pentndecdchordi t f^mitonio dmplius qudm tenfus fit 
profldmbdnomenos pentadccdchordi s.cui tetrdchoria per qudrtdm^ quinmm, CT 
fcxmm /)«b^ ut prius codpto.eritque per diffinitionem:t pentndecdchordumhyfo^ 
iydtj modi. C7 pr^fldrnhdnommon u uno tono extendo amplius qudm bypolydij: 
cuic^terdsuoccsfiio ordine fiquentes in didtomco gencre codpto.eritque cocentus 
pciadecdchordi uiper diffinitiorx dori^,crfi dmplius extedo x uno tono:erit vetn 
dccdchordi x cdntus phrypus. ^fiy femitvnio minorcierit eius contentus lydius. 

1 
1 

I 



p 



M 



At X 'mfUus extenfi irw tono- fa tcucentus myxciyiius.f,ci}i faOum frr ^~ 

l^lZu6o: ca eric fccundx ad fccuodam «c rem. «d cmum, & 
cuiud.brt rotias adro»mfirnih«,cade(rKg babirudo. 



JylopCyW rrrij_i I i i m rrr^ 



7 J f~w -X A I. c d g f g 



«,L,.,«mrt.»«t^r/^.Bt4Jr.4t44Jamo«/b'af«..;lr//«^^ 

CTorui ordod.atonicuspcntadccachord.hypophrygn mod.:ton.m 15 
hvpodorium vnius acummc toni fupcrar. & torus hypolydms ciidem 
tr.Untonio.r.ngul* quoqac doriifingulas hypodon, d>«cllaron_con- 
fonanna. torus vcrophrygiusconfonantia dupc.uc lyd.us autcdia. 



pence atq, fcmitonio.& myxolydius di.pentc atquc Icfquirono. 



A fc c a f r j: ^ ^ k I fy» » 




Z M>x£)'>Jr«i 



6 c d c I ^ b I K r m w 



CNim rrorMnbjnomcnoshyf^of^hryiif.tomfuf^rrat ac mtftc projljmb^nomcnon 
hypodonj.cr^o fcr ^r^ccdtntem totus hyyophry^ius ordo totum hyf^odorium or. 
dinem tomfupcrAtacumwe^cr quufroUmhjncmcnoshyf>olydiifu^rratdCumf 
mfimitonu minoris hy^o^hyy^um:crioidcm fuicr^t^umincpolUmbanomeno 



IITI 4) 

hypoJhmm trijimitmioAgiturferfr^ecedeHtemttotus hyf^Jydius ^dototum hy 
fodoriwm ordinew tripmitmojiffreuaditdcutiorfpd Crgww dcrius tonidcutic 
tuncU hy^lydium:crgo dorius duohus tonis cr fimitomo acutior efl bypajorw.fr- 
|o ferfexmm tertij.eo acutior fR confindtitid diateffaron.qudre ftrfr^ec^dtntem 
iinguU dorijfiniulis hyj^or^di^cffdroncovfonantia finant dcutiores.fjx f^y^ 
pusdddit wmm in dcutiunt ibno\\ptwt wtus fhrypus totohyfodmOfdidfente 
confindntid moduldtur dcutior^o' lyditis fhrygio addit femiwnmfn minui^trnU 
xolydiUi lydio tomm:igttuY lydius hyfodoriodiapcnte^pfmtoniOy&niixolydi' 
us didpcnt€ ^pfquitonQJonahit acutior. quod totum eft prc^ofttwrn. 

^^ CHypolydius diaronicus:hypophry giam diatonicum {cmiton!o,& do 
rius trifcmitonio,phrygius diaccflaron,Iydiu$ diatcflaron&femitonio 
myxolydiu$ diatcfliroii confonantia, atquc fefquitono fupcreuadit a» 
cucior.dc dorius bypolydio tono,phry gius ditono, lydius diaccflaron, 
dc my xoly dius diapentc.phry gius autcm ad dorium fonat ronum,lydi 
us cnfcmitonium& myxolydius diatcflaronjlydius phrygio femito- 
fuuni,& myxolydius Icfquitonum.myxolydiusautemlydioitonum. 

€k^c ut fY^ccdcns uel quam faciUime monflrdbitur. 

ij CScptcm modoschromaticc modulationisconftitucrc, 

Afc cdefghklmnofq 

KHypodorn*s\ i I I I III I I I j j I I 1^7' _ 1 

5 Hyyoph ry. \1ormy | | | | I I I M 1 II I \\qChr0 



Tnyfol^iT Ts^rnito. A | | | I I I M I I I j H j^^^ 

VDon^'^^ l^potomc Aj J \ \ WJ J I I I fl I ^^''^' 

x^y^i JTrjCem itonium A^j 1 j | |J I I II l l I H '^"^ 

Yj^ydius I semitonium ^Ij J_ I I UlJ.Lll"! I **'• 

Z Myxof>diMi I ATOfowf A | | (X M I L I J LU J- 

' a6 cd € f^h k I mno fq 



€f4cioYvcntad(Cdchordum chromdtkum ferdccimdnj.mJecimamy O-d^^decu 
mm buiuVcr /imilirrr/f x ^\xd[d\\ctt Syt,u,x,y,i.fdcioqu€ pntddtc dehor dtwi s 
tonoacutius r.ef t.^^tnxjidccdchordwm'fcmwnioacutius s.u um:dfetomes inter 
uaUo acutius tO' xitYifemitvniodmplius quamuy:femmiodmflwsqu 
:^.afotomctrdnfc{ndireyMcoer^ofej^tmmodos chromatict moMdtiomsejje 



M 

cribu^s. 9m rait hyfpiaruis.s frrSffimtknm hy^cfhfypusdrcmdtku's . i 
byfolydius.u doriusa fbry^.y lyiiHs.':imyxolydius. 

CQuo pafto finguli cuUibct inter fc chromatid modi rcfpondcantro* ^ 

fteodcre* . . , , , 

CDi^ pff ffotedentem hy?^phrypus nt frypoJarrd (hromdtko: mo CT ^P^ 
lydif^s db hy^fhrygw:fimicom0,diftat igitwrhypolydms ab hypodorioidcutioY tri 
fimito^O'Cr dmus Jb bypolydio i^dtAp^mtA^twt dormi 4b hyj^odorio. r^m^ 
tus ^ dimno.fhrypHS dttUm db dariotrifirmtomo^iptitr fhfygiit^hyfodom ft 
mtmrtur tribus torn c fcf^amo^hoc fjl t^tms amfimiMtut didfcnte intnudlU* 
Cr lydius 4 ^gbftmiamo.tgkurly^s A hy^mo dUfmeatquefmitot^ 
fftyxdydius db Qiu difidt dfiHomc.iptur myxoiydius A hypodmo didfottt cS^ 
fondntUdifidtdcutWTfdtqutt9H0.uhocpdaodereUquis ex Pt^edente ffffdcu 
lis cji ofh^toM hyfolydius db hy^ibrypo dijlstfimimicJmus tono.fhrygfus 
J^jtelftroHAydiHsdutcjfaron cr P^^^^^m^y^^^ dUpcnte. Dcrius d hyfdy 
djo difidt dfoa>me.fhrypHS ditonoylydius diattffdfo^myxolydius ttvpn^-VhtygU 
usddorio trifimivnioJydius tonocr duobus fimitomis minoYibus.m^oly^s ue 
to dutefpcron,Ly^s d pbrygia fim^mo.cr ntyxoly^us wno.dijldt untcm myX0 
lydius ut utm qitoque diihtm ejlidlydio mdwrc jermcomo. ftcqu€ conJhru^Hm tfi 
^rofofitum, 

CScptcra iridcm cnarmpnicos modos ordinary ^ 

Abcdcfghklmnofq 



K Hypodofius\ I I I I I I 11 I Ml I 1 I 1^7 

f Hypovhry.\ronus I I I I I 1 M I 1 ] I I I l\q^n^r 

rnypolydi. \Ditlts a M I 1 I I I TTj I I I 11 \q rno 

XTHnius iDirfts A | | | | | | | || J | | f ]T? >^' 

X i>,>rygius I Diton us A | ] | | | | | | 11 I I I I I \ \ H '^ , 

Y Lydius^ r ^fs ^ A I 1 1 1 1 rn ixriTi 1 1 ^^ 

xH^ohii^l^ts ^ ~]~l'r MM T riMLI_U!* 

A b c d e f^h k I m no pq 



i[iltar,f^t,u,Xyy,^/eftcmp€Htddecachord4^0'fitperdecimdmqufHt4m huiusr 
pentddtcdchordum cnarmomc urn :ijt tend pr^^dmbanomfnon pcnradecachordi i 
no wnodmplius pentddecjchordo r.O" t diefi ampUus qudm i.cr « dUjldmpl 



It us 



3u4m f.cr X ditono ampUtis qusm u. cr> dmplius dufi x. CTli amplius itidcm 
left ({Udm y.erif^ ergoficundum ucumen intcnftfex pojl primum pcntndccdchor 

dd 



nil 44 

dd:fnim toH0,d<iHi€ iudlutikfibus 0* iitono^iemm iuabus iiepbus.qui qui* 
dcm intendendi modusienarmwkc m&duUtiom feculans frcfrwffit habitur.Jut 
ipturfiftemfentadecMchordd r,fjty «, x, >, q^: /eftm cHAmomce moduUtims 
wodcs coutitt€iitid.quod dtmojjRrari po^fxtum €rat. 

GSingulorum cnarmonicorum modorumud quemlibet habltudines 
^ dcmoaftrare, 

tiHuius ex ff^cedenti demonftrdtiocUra effe fQteft.Vrhno hyfofhrys^ij ab hyfo- 
dorio ^ftanthimieffe wnunt^hyfolydiptomim cr ^fimJorifijifquiwnunt.fhYyiij 

dwfr»trJ>dij:dMpcjire 4cdJr/iw.w:yxoJ>dij;dwpoifr cr/^^^ ^y 

fQlydiumdiftdre db hyfofhrypo.diefudomm:finMonto.fhrypum:dute^^^ 

duimidUteffdron ^ dufi.myxidydiumididUllaron CT fi^f*^^^^^'^^^^^^^^^^'*^ ^ 
hf>^^'dUfufhryffum:dUon0 ct dUfulydmM:diateffdY<m.wyxvlydium: didtef 
fitrono'diffi'QJi^Ttofbryffmn a dcnoiditDnQ.lydiuMiditono^dUfumyxolydi* 
ufM:diAUffdrQn.Quintoly^m d pbry^io.-Ar/i.cr m>xo(ydww;/?wwtt)wio. sexto 
v^xoiy^um diftare diefid lyc^o. scftem enm modes O" ^^^ pi»rei ddiccit (rif- 
cQTum duthoTUds Vythdg0ricorum,Yt emm numnus 4 monadt dd dcudrimn ufr 
queudTiHScrcfcensfrogredituriWoxuerofiquens dcndYius uoiimtis uiccm obtu 
net primamquc €xfitcdtt0titdt€m dufdtm indmduamonddisconjors^ CT^fiut 
lus^qui dd centcndjium ufquc rurfus noutrtdtut frogreffione fi txtendit^ reU^s 
tandem in tcrtidm unitatem:' itdquoqueuocum dilJimtUtudo dcuarietds ex qui^ 
bus wftar caleflis hdtmonia concentus humdnifmfdique formdntur ad oilona- 
rium ufque fidtpt. funtquefef tern continue UQces inter ft udfue: quibus fucctdens 
o^onarixuoc'ts fknitudo^frimuswimnuMetdlis iubus ffmique teffera oilona* 
rius) .id frimam rurfus fonat ut eadem ,^ad earn ftfe habcm ferinde dc dend-- 
rius dd unitdt€m,tt hisec o^onarid feries in omni tncduiathnis genere ftcratdfro* 
cedit'MtcotinuioildUQquoque loco oHduutn fcrftmiUmftbi^fome eundim fo* 
nus offenddt fonum.itd ut ex duobus natiud quddamjconcordique ajfinitdte: iam 
unumfbnum cr non tnultos farere uidedntur.ufque ddeo enim fe miJcent:ZT w» 
tuafe iunguut^coj^uldntqueamicitid*ithdrump{temuocmnquaegrauiffimd tdr* 
diflimdque cfliSaturno dcbetur.froximdiioui.tmia M^irtLqudrtaiPhabcqumta 
\inmpxt*i:^Urcurw,feftimdU€ro tdrum dcutiffimd^concitdtiffitnaque: l«»^« 
cd^ua amemiruyfum reuolultur dd Sdturnunt.nondidd louem decinididd Mdrtem 
t7 hoc fdlbo co>if€quint€s:ut pif tens uoluit dntiquttds.ttreutrd totius mtiuerfi 
ha7tnomdfeft(9^dywcdfUta efi.^hacy in ccelo c^eUftcyin his autiinftrioribus 
corporca^jenfibiUmque teferathdrmonia.fedh^ctndgi flenius difcutiatlU licet 
cQ^nofcerccurhyferviyxolydiuhJ ptb,medii baud vtultuuemat acccmodddus. 

L iUj Udm 



1j 



r 



i«p 



tM fMtgr.qudre mji air hyf oiorh^fthn^ m^ cm^famm uarim futmklHS 
tftcr^i^fi^ n^yx^lyiii^s intmincturtnfimixfmoin dmwi^ CT *litt»fO w 

modi quasddigcere recentiorcs ^t diuus Gregmus: £ hoc nMormm dff^qmtsH 
receditm.O' fl^dqutaluqtfa fojlcmrcs muftd inculcduermnt. frqu^dbiiUs 
fadUreqwfds.O' n^fhaquoqaettrnfejldtermfKum moinLmen^quc omikm 
coiKUuntijm ad celeriatcm qHOHidf^pr^cifittrnqMc Uuimtcm mucerc cojmwj - 
tunmoicflAm^pmtm firumqutdc icc^4m c^ncenimim imderdthnem p«rp4rtt 
dttcndentcs d modctdtionc tnm di^ {kht woii. ^dtwm item Mtnientes fr^ci 
rmficcs honefldtis^dKitdtifquedxtuiqiid dmentes drreftiti9f[^foluAdnt,frmtd 
Hs inducebdHtyftros horrmmm mar€s(ut dm rhrdcius Orflueus)dd m^ct^s 
^uiftmtis cdiUmrciiocAdHt,^ ex finftbiiHm hsrmMXddd c^tkftis hmmmM 
defhicrium cdj^ds dmrndstrnftqudm fui idm mem<frcs cxtfij ubcrfim fluentAi^s o 
culis eit^dbdnt.h^ cnmin^d^rttm dccomm^d mcdi^criutte TyAdgor^ difapuH 
fxcicbantMH enhn is inter homines m9defHor:adus omnis mc^tts curfis ukd^f 
Ur,ne<^ is emus fracefswmmn hqueU ^JemimnUtdU intefUgentidm. ita 
quoqut neque ij modeftiores moii:qKi nmidfuififtmdnthtquaft in uenercA choi 
red Ufcimentes frMcrfygumt dudUumhdc enm ik cMt/d Vythdgorea phoU md* 
cs cbromais modoi re^udiMuSio^ S^tr^dt^tfolenm decretp Timoth^um mikfi- 
urn increfuerunt u^menterdi^iDnicos cancetitus fctius dfprobdntes, qui ft dftc 
modertmufiuirtiitis fr^pfirunt modtjHdm.mteftnnmfffii tdrJ^dte ceu tcrpore 
quoddm fdftidirmrindniutd nirm4 cderitds moUictcma^tdrtddrnPra fi fertinho* 
n^am.meiitim enm neque eeleriostt fr^etfs nefte ^fiuud t^itdte pfgrw: I^ 
ddbOe wrtutfyue emuLm. qudm Qmnis ^euf frobMOtt ynbdtmdqne e^. & (^ 
qudm ffwfidmfdi tdnqudm tnoderdtioms ofdmiqtueddmteHeregiiams fcrAu 
cere dAet^^CT dd diuind mentes nojhds iugiter rdpereiirf'^^^^M eruntujui hoc 
fint cr rmfteen cr omnem nrnnidndm fhUojofhi^m qu^terintjuque tdUms de* 
elfe filet caleflis fdHordtqu^fr^tdium.qHidinemfiKtisfdcidntmferiiqv^^ nul 
ios dd quos nofhd h^c modHidticnum eUmentd feruenennt futures deftdefdmus, 
quin eis omms hsrmonU wtde decus ^AkdiUt^fUmus ,nojbri(fit mduros memo- 
res, 

FI N I S.